Calculates design matrices to fit a natural polynomial or trigonometric L-spline as a linear mixed model (S.J. Welham).
Options
KMETHOD = string token |
Method for constructing the set of knots (equal, quantile, given); default equa |
|---|---|
NSEGMENTS = scalar |
Specifies the number of segments between boundaries; default * obtains a value automatically |
INKNOTS = variate |
Provides the set of internal knots when KMETHOD=given |
CORE = string token |
The form of core function to use; (cossin, intcossin, lincossin, intercept, linear, quadratic) default linc |
PERIOD = scalar |
Defines the period for trigonometric functions (not required for polynomial splines) |
LOWER = scalar |
Specifies the lower boundary when KMETHOD=equal; default takes the minimum value in X |
UPPER = scalar |
Specifies the upper boundary when KMETHOD=equal; default takes the maximum value in X |
ORTHOGONALIZETO = variate |
Variate to use to get an orthogonalized basis; default * i.e. orthogonalization with respect to X |
SCALING = scalar |
Scaling of the XRANDOM terms (automatic, none); default auto |
Parameters
X = variates |
The explanatory variate for which the spline values are required |
|---|---|
XFIXED = matrices |
Saves the design matrix to define the fixed terms (excluding the constant) for fitting the L-spline |
XRANDOM = matrices |
Saves the design matrix to define the random terms for fitting the L-spline |
KNOTS = variates |
Saves the internal knots and boundaries used to form the basis for the spline |
PX = variates |
Specifies x-values at which predictions are required |
PFIXED = matrices |
Saves the design matrix for the fixed terms (excluding the constant) for the spline at the prediction points |
PRANDOM = matrices |
Saves the design matrix for the random terms for the spline at the prediction points |
Description
This procedure generates the fixed and random terms required to fit a polynomial or trigonometric L-spline as a linear mixed model, using REML estimation of the smoothing parameter (Welham et al., 2006). The explanatory variate values at which the spline is to be calculated are specified, in a variate, by the X parameter.
The KMETHOD option specifies how to choose the set of knots for the penalized spline, using settings:
equal |
splits the range of X into segments of equal length (default), |
|---|---|
quantiles |
defines the set of knots in terms of equally-spaced quantiles of X, |
given |
indicates that the knots will be supplied, in a variate, by the INKNOTS option. |
The number of segments or quantiles for the equal and quantile settings is specified by the NSEGMENTS option. If this is unset, the number is determined automatically as
min([p/4], 35) + 1
(Ruppert 2002) where p is the number of unique values of the variate X and [r] denotes the integer part of the number r. The lower and upper boundaries for equal segments are specified by the LOWER and UPPER options, respectively, taking the minimum and maximum values of X as their defaults. The set of knots used to form the spline basis can be saved using the KNOTS parameter.
The form of the core functions used to generate the L-spline, i.e. the form that will remain unpenalized, is specified using the CORE option. The settings cossin, intcossin and lincossin define trigonometric L-splines, with a (known) period defined by the PERIOD option. These have the following forms:
cossin |
a1 cos(kx) + a2 sin(kx) |
|---|---|
intcossin |
a0 + a1 cos(kx) + a2 sin(kx) |
| lincossin | a0 + a1 + a2 cos(kx) + a3 sin(kx) |
where
k = 2 π / PERIOD
and the ai are unknown coefficients.
The settings intercept, linear and quadratic define polynomial splines, as follows:
intercept |
natural linear spline with a single constant core function, |
|---|---|
linear |
natural cubic spline with linear core functions, and |
quadratic |
natural quintic spline with quadratic core functions. |
If the distinct values of X are used as knots, the function will produce a basis for a polynomial smoothing spline; with fewer knots, the spline is a low-rank approximation equivalent to the O-splines described by Ormerod & Wand (2008). A different parameterization of the basis is used here, but the fitted spline will be the same.
The ORTHOGONALIZETO option specifies a variate to use in orthogonalization. The set of random spline terms will then be orthogonal to the fixed terms when evaluated at the specified values. For most data sets, it is recommended to set ORTHOGONALIZETO to the variate X (the default). The random terms will then be orthogonal to the fixed terms, and fitted values corresponding to the fixed model will represent the whole of the polynomial trend in the fitted spline. For very large data sets, this calculation can be onerous and can be approximated by making the two bases orthogonal at the knots. No orthogonalization is carried out if ORTHOGONALIZETO is set to a scalar value (eg. ORTHOGONALIZETO=0).
The L-spline terms are saved as two matrices. The terms required to be fitted as fixed terms can be saved using the XFIXED parameter. For CORE=cossin, the constant should be omitted from the model by setting option CONSTANT=omit in VCOMPONENTS. For CORE=intercept, there are no additional fixed terms, and so XFIXED is ignored. The terms to be fitted as random can be saved using the XRANDOM parameter.
The random terms can be scaled so that, for a random spline matrix Z,
TRACE(Z *+ T(Z)) = NROWS(Z)
This ensures that the average contribution of Z to the variance of an observation is equal to one, and hence the overall contribution from the term is equal to the spline variance component. This is highly recommended as it removes computational instabilities due to the intrinsic scale of the matrix K (see Method), and improves interpretability of the spline variance component. This scaling is imposed by default, but can be avoided by setting option SCALING=none. For L-splines, use of some scaling is strongly recommended, as the unscaled matrix can be so large or small that the associated variance component appears aliased and cannot be estimated.
The L-spline terms required for prediction can be saved using the PXFIXED and PXRANDOM parameters. The PX parameter defines the set of x-values at which the predictions are to be made.
Options: KMETHOD, NSEGMENTS, INKNOTS, CORE, PERIOD, LOWER, UPPER, ORTHOGONALIZETO, SCALING.
Parameters: X, XFIXED, XRANDOM, KNOTS, PX, PXFIXED, PXRANDOM.
Method
The L-spline with core functions ∑ τi di(x) and r knots, evaluated on variate X, minimizes the penalized sum of squares
(y – X τ – Kx c)ʹ R-1 (y – X τ – Kx c) + λ cʹ K c
where
X is a design matrix containing the core functions evaluated at x, with associated unknown parameters τ;
Kx is a design matrix containing r L-spline basis functions evaluated at the knots, with associated unknown effects c; and
K is an r × r symmetric matrix of L-spline basis functions evaluated at the knots (for details see references in Welham et al. 2006)
The matrix K can be transformed to full rank as
H = Cʹ K C
where the matrix C contains the eigenvectors of XXʹ corresponding to zero eigenvalues, with corresponding transformation of the matrix functions as
Ku = Kx C
This is translated to a set of independent random effects via post-multiplication by H-1/2.
The penalized sum of squares is reformulated as the estimating equations from a mixed model of the form
y = X τ + Z u + e
where
Z = Kx C Hm-1/2
u is a set of r independently and identically distributed random effects with var(u) = σs2 I
e is a vector of residual errors with var(e) = σ2 R
Fitting this mixed model with known λ set equal to σ2/σs2 produces estimates that minimize the penalized sum of squares. In addition, we can estimate the smoothing parameter using REML via the variance component σs2. This can be generalized straightforwardly to mixed models with additional fixed and random terms.
The implementation in this procedure allows the random design matrix to be orthogonalized with respect to the fixed design matrix at a given variate. For orthogonalization with respect to the variate X, this is achieved by using random design matrix
Z* = (I – X(XʹX)-1Xʹ)Z
The entirety of the polynomial trend is then captured by the fixed model. Orthogonalization with respect to a variate t is calculated as
Z* = Z – X(TʹT)-1Tʹ Kt C Hm-1/2
where T is a matrix holding the core functions evaluated at t, and Kt is a matrix of L-spline basis functions evaluated at t. No orthogonalization is carried out if ORTHOGONALIZETO is set to a scalar value (e.g. ORTHOGONALIZETO=0).
When the random matrix is scaled so that trace(Z*Z*ʹ) is equal to the number of row of Z*,
the average contribution of the spline term to the variance of each unit (σs2 × diag(Z*Z*ʹ)) is equal to σs2. This makes the spline variance component value directly comparable with the residual variance.
Note that the constant function is not included in the fixed design matrix generated by PENSPLINE, as this term is added automatically to the linear mixed model by the default option setting, CONSTANT=estimate, in the VCOMPONENTS statement.
The design matrices for use in prediction are calculated by evaluating the same set of basis functions at the predict points.
Action with RESTRICT
Input structures must not be restricted.
References
Ruppert, D. (2002). Selecting the number of knots for penalised splines. Computational & Graphical Statistics, 11, 735-757.
Wand, M.P. & Ormerod, J.T. (2008). On semiparametric regression with O’Sullivan penalised splines. Australian & New Zealand Journal of Statistics, 50, 179-198.
Welham, S.J., Cullis, B.R., Kenward, M.G., Thompson, R. (2006). The analysis of longitudinal data using mixed model L-splines. Biometrics, 62, 392-401.
See also
Directive: VCOMPONENTS.
Procedures: SPLINE, NCSPLINE, PENSPLINE, PSPLINE, RADIALSPLINE, TENSORSPLINE.
Function: SSPLINE.
Commands for: Calculations and manipulation, Regression analysis, REML analysis of linear mixed models.
Example
CAPTION 'LSPLINE examples'; STYLE=meta
" generate data "
CALCULATE [SEED=23826] x = GRUNIFORM(30; 0; 100)
& true = 10 - 0.005*(x-27)**2 - 2*SIN(2*C('pi')*x/40)
& [SEED=0] y = true + GRNORMAL(NVAL(x); 0; 1)
" pens for plotting lines "
PEN 2,3; METH=LINE; SYMBOL=0; LINE=1
" plot data "
DGRAPH [TITLE='Observed data'] y; x
" generate L-spline basis for core functions 1, x, cos(kx), sin(kx)
where k (=2*c('pi')/40) gives period 40 for trigonometric functions
and design matrices to predict at 1 unit intervals from 0 to 100 "
VARIATE [VALUES=0...100] xpred
LSPLINE [NSEGMENTS=10; LOWER=0; UPPER=100; PERIOD=40]\
x; XFIXED=X; XRANDOM=Z; KNOTS=knots;\
PX=xpred; PFIXED=PX; PRANDOM=PZ
" plot random basis functions (held in columns of Z) "
DGRAPH [TITLE='Random basis functions for L-spline'; KEY=0]\
Z$[*;1...7]; x; PEN=2
" check set of knots used "
PRINT knots
" fit L-spline as a linear mixed model "
VCOMPONENTS [FIXED=X] RANDOM=Z; CONSTRAINT=positive
REML y
" save fitted values "
VKEEP [FITTED=fv]
" plot data with fitted values - curve not smooth due to gaps between
observations and linear interpolation on plot "
DGRAPH ['Data with interpolated L-spline'; KEY=0] y,fv; x; PEN=1,2
" predict at 1-unit intervals (using PX and PZ generated earlier) "
VPREDICT [PREDICT=Tp] X,Z; LEVELS=PX,PZ; PARALLEL=*,X
" extract and plot predicted curve with data "
VTABLE Tp; VARIATE=Vp
FRAME [GRID=xy] 1
XAXIS 1; MARKS=knots
DGRAPH ['Data with fitted L-spline'; KEY=0] Vp,y; xpred,x; PEN=2,1