bigssp {bigsplines} | R Documentation |
Fits Smoothing Splines with Parametric Effects
Description
Given a real-valued response vector \mathbf{y}=\{y_{i}\}_{n\times1}
, a semiparametric regression model has the form
y_{i}= \eta(\mathbf{x}_{i}) + \sum_{j=1}^{t}b_{j}z_{ij} + e_{i}
where y_{i}
is the i
-th observation's respone, \mathbf{x}_{i}=(x_{i1},\ldots,x_{ip})
is the i
-th observation's nonparametric predictor vector, \eta
is an unknown smooth function relating the response and nonparametric predictors, \mathbf{z}_{i}=(z_{i1},\ldots,z_{it})
is the i
-th observation's parametric predictor vector, and e_{i}\sim\mathrm{N}(0,\sigma^{2})
is iid Gaussian error. Function can fit both additive and interactive non/parametric effects, and allows for 2-way and 3-way interactions between nonparametric and parametric effects (see Details and Examples).
Usage
bigssp(formula,data=NULL,type=NULL,nknots=NULL,rparm=NA,
lambdas=NULL,skip.iter=TRUE,se.fit=FALSE,rseed=1234,
gcvopts=NULL,knotcheck=TRUE,thetas=NULL,weights=NULL,
random=NULL,remlalg=c("FS","NR","EM","none"),remliter=500,
remltol=10^-4,remltau=NULL)
Arguments
formula |
An object of class " |
data |
Optional data frame, list, or environment containing the variables in |
type |
List of smoothing spline types for predictors in |
nknots |
Two possible options: (a) scalar giving total number of random knots to sample, or (b) vector indexing which rows of |
rparm |
List of rounding parameters for each predictor. See Details. |
lambdas |
Vector of global smoothing parameters to try. Default |
skip.iter |
Logical indicating whether to skip the iterative smoothing parameter update. Using |
se.fit |
Logical indicating if the standard errors of the fitted values should be estimated. |
rseed |
Random seed for knot sampling. Input is ignored if |
gcvopts |
Control parameters for optimization. List with 3 elements: (a) |
knotcheck |
If |
thetas |
List of initial smoothing parameters for each predictor subspace. See Details. |
weights |
Vector of positive weights for fitting (default is vector of ones). |
random |
Adds random effects to model (see Random Effects section). |
remlalg |
REML algorithm for estimating variance components (see Random Effects section). Input is ignored if |
remliter |
Maximum number of iterations for REML estimation of variance components. Input is ignored if |
remltol |
Convergence tolerance for REML estimation of variance components. Input is ignored if |
remltau |
Initial estimate of variance parameters for REML estimation of variance components. Input is ignored if |
Details
The formula
syntax is similar to that used in lm
and many other R regression functions. Use y~x
to predict the response y
from the predictor x
. Use y~x1+x2
to fit an additive model of the predictors x1
and x2
, and use y~x1*x2
to fit an interaction model. The syntax y~x1*x2
includes the interaction and main effects, whereas the syntax y~x1:x2
only includes the interaction. See Computational Details for specifics about how non/parametric effects are estimated.
See bigspline
for definitions of type="cub"
, type="cub0"
, and type="per"
splines, which can handle one-dimensional predictors. See Appendix of Helwig and Ma (2015) for information about type="tps"
and type="nom"
splines. Note that type="tps"
can handle one-, two-, or three-dimensional predictors. I recommend using type="cub"
if the predictor scores have no extreme outliers; when outliers are present, type="tps"
may produce a better result.
Using the rounding parameter input rparm
can greatly speed-up and stabilize the fitting for large samples. For typical cases, I recommend using rparm=0.01
for cubic and periodic splines, but smaller rounding parameters may be needed for particularly jagged functions. For thin-plate splines, the data are NOT transformed to the interval [0,1] before fitting, so the rounding parameter should be on the raw data scale. Also, for type="tps"
you can enter one rounding parameter for each predictor dimension. Use rparm=1
for ordinal and nominal splines.
Value
fitted.values |
Vector of fitted values corresponding to the original data points in |
se.fit |
Vector of standard errors of |
yvar |
Response vector. |
xvars |
List of predictors. |
type |
Type of smoothing spline that was used for each predictor. |
yunique |
Mean of |
xunique |
Unique rows of |
sigma |
Estimated error standard deviation, i.e., |
ndf |
Data frame with two elements: |
info |
Model fit information: vector containing the GCV, multiple R-squared, AIC, and BIC of fit model (assuming Gaussian error). |
modelspec |
List containing specifics of fit model (needed for prediction). |
converged |
Convergence status: |
tnames |
Names of the terms in model. |
random |
Random effects formula (same as input). |
tau |
Variance parameters such that |
blup |
Best linear unbiased predictors (if |
call |
Called model in input |
Warnings
Cubic and cubic periodic splines transform the predictor to the interval [0,1] before fitting.
When using rounding parameters, output fitted.values
corresponds to unique rounded predictor scores in output xunique
. Use predict.bigssp
function to get fitted values for full yvar
vector.
Computational Details
To estimate \eta
I minimize the penalized least-squares functional
\frac{1}{n}\sum_{i=1}^{n}\left(y_{i} - \eta(\mathbf{x}_{i}) - \textstyle\sum_{j=1}^{t}b_{j}z_{ij} \right)^{2} + \lambda J(\eta)
where J(\cdot)
is a nonnegative penalty functional quantifying the roughness of \eta
and \lambda>0
is a smoothing parameter controlling the trade-off between fitting and smoothing the data. Note that for p>1
nonparametric predictors, there are additional \theta_{k}
smoothing parameters embedded in J
.
The penalized least squares functioncal can be rewritten as
\|\mathbf{y} - \mathbf{K}\mathbf{d} - \mathbf{J}_{\theta}\mathbf{c}\|^{2} + n\lambda\mathbf{c}'\mathbf{Q}_{\theta}\mathbf{c}
where \mathbf{K}=\{\phi(x_{i}),\mathbf{z}_{i}\}_{n \times m}
is the parametric space basis function matrix, \mathbf{J}_{\theta}=\sum_{k=1}^{s}\theta_{k}\mathbf{J}_{k}
with \mathbf{J}_{k}=\{\rho_{k}(\mathbf{x}_{i},\mathbf{x}_{h}^{*})\}_{n \times q}
denoting the k
-th contrast space basis funciton matrix, \mathbf{Q}_{\theta}=\sum_{k=1}^{s}\theta_{k}\mathbf{Q}_{k}
with \mathbf{Q}_{k}=\{\rho_{k}(\mathbf{x}_{g}^{*},\mathbf{x}_{h}^{*})\}_{q \times q}
denoting the k
-th penalty matrix, and \mathbf{d}=(d_{0},\ldots,d_{m})'
and \mathbf{c}=(c_{1},\ldots,c_{q})'
are the unknown basis function coefficients. The optimal smoothing parameters are chosen by minimizing the GCV score (see bigspline
).
Note that this function uses the classic smoothing spline parameterization (see Gu, 2013), so there is more than one smoothing parameter per predictor (if interactions are included in the model). To evaluate the GCV score, this function uses the improved (scalable) SSA algorithm discussed in Helwig (2013) and Helwig and Ma (2015).
Skip Iteration
For p>1
predictors, initial values for the \theta_{k}
parameters are estimated using Algorithm 3.2 described in Gu and Wahba (1991).
Default use of this function (skip.iter=TRUE
) fixes the \theta_{k}
parameters afer the smart start, and then finds the global smoothing parameter \lambda
(among the input lambdas
) that minimizes the GCV score. This approach typically produces a solution very similar to the more optimal solution using skip.iter=FALSE
.
Setting skip.iter=FALSE
uses the same smart starting algorithm as setting skip.iter=TRUE
. However, instead of fixing the \theta_{k}
parameters afer the smart start, using skip.iter=FALSE
iterates between estimating the optimal \lambda
and the optimal \theta_{k}
parameters. The R function nlm
is used to minimize the GCV score with respect to the \theta_{k}
parameters, which can be time consuming for models with many predictors.
Random Effects
The input random
adds random effects to the model assuming a variance components structure. Both nested and crossed random effects are supported. In all cases, the random effects are assumed to be indepedent zero-mean Gaussian variables with the variance depending on group membership.
Random effects are distinguished by vertical bars ("|"), which separate expressions for design matrices (left) from group factors (right). For example, the syntax ~1|group
includes a random intercept for each level of group
, whereas the syntax ~1+x|group
includes both a random intercept and a random slope for each level of group
. For crossed random effects, parentheses are needed to distinguish different terms, e.g., ~(1|group1)+(1|group2)
includes a random intercept for each level of group1
and a random intercept for each level of group2
, where both group1
and group2
are factors. For nested random effects, the syntax ~group|subject
can be used, where both group
and subject
are factors such that the levels of subject
are nested within those of group
.
The input remlalg
determines the REML algorithm used to estimate the variance components. Setting remlalg="FS"
uses a Fisher Scoring algorithm (default). Setting remlalg="NR"
uses a Newton-Raphson algorithm. Setting remlalg="EM"
uses an Expectation Maximization algorithm. Use remlalg="none"
to fit a model with known variance components (entered through remltau
).
The input remliter
sets the maximum number of iterations for the REML estimation. The input remltol
sets the convergence tolerance for the REML estimation, which is determined via relative change in the REML log-likelihood. The input remltau
sets the initial estimates of variance parameters; default is remltau = rep(1,ntau)
where ntau
is the number of variance components.
Note
The spline is estimated using penalized least-squares, which does not require the Gaussian error assumption. However, the spline inference information (e.g., standard errors and fit information) requires the Gaussian error assumption.
Author(s)
Nathaniel E. Helwig <helwig@umn.edu>
References
Gu, C. (2013). Smoothing spline ANOVA models, 2nd edition. New York: Springer.
Gu, C. and Wahba, G. (1991). Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method. SIAM Journal on Scientific and Statistical Computing, 12, 383-398.
Helwig, N. E. (2013). Fast and stable smoothing spline analysis of variance models for large samples with applications to electroencephalography data analysis. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign.
Helwig, N. E. (2016). Efficient estimation of variance components in nonparametric mixed-effects models with large samples. Statistics and Computing, 26, 1319-1336.
Helwig, N. E. (2017). Regression with ordered predictors via ordinal smoothing splines. Frontiers in Applied Mathematics and Statistics, 3(15), 1-13.
Helwig, N. E. and Ma, P. (2015). Fast and stable multiple smoothing parameter selection in smoothing spline analysis of variance models with large samples. Journal of Computational and Graphical Statistics, 24, 715-732.
Helwig, N. E. and Ma, P. (2016). Smoothing spline ANOVA for super-large samples: Scalable computation via rounding parameters. Statistics and Its Interface, 9, 433-444.
Examples
########## EXAMPLE ##########
# function with four continuous predictors
set.seed(773)
myfun <- function(x1v,x2v,x3v,x4v){
sin(2*pi*x1v) + log(x2v+.1) + x3v*cos(pi*(x4v))
}
x1v <- runif(500)
x2v <- runif(500)
x3v <- runif(500)
x4v <- runif(500)
y <- myfun(x1v,x2v,x3v,x4v) + rnorm(500)
# fit cubic spline model with x3v*x4v interaction and x3v as "cub"
# (includes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v*x4v,type=list(x1v="cub",x2v="cub",x3v="cub",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500
# fit cubic spline model with x3v*x4v interaction and x3v as "cub0"
# (includes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v*x4v,type=list(x1v="cub",x2v="cub",x3v="cub0",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500
# fit model with x3v*x4v interaction treating x3v as parametric effect
# (includes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v*x4v,type=list(x1v="cub",x2v="cub",x3v="prm",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500
# fit cubic spline model with x3v:x4v interaction and x3v as "cub"
# (excludes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v:x4v,type=list(x1v="cub",x2v="cub",x3v="cub",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500
# fit cubic spline model with x3v:x4v interaction and x3v as "cub0"
# (excludes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v:x4v,type=list(x1v="cub",x2v="cub",x3v="cub0",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500
# fit model with x3v:x4v interaction treating x3v as parametric effect
# (excludes x3v and x4v main effects)
cubmod <- bigssp(y~x1v+x2v+x3v:x4v,type=list(x1v="cub",x2v="cub",x3v="prm",x4v="cub"),nknots=50)
crossprod( myfun(x1v,x2v,x3v,x4v) - cubmod$fitted.values )/500