gsp: General regression splines with variable degrees and...
In gmonette/spida: Collection of miscellaneous functions for mixed models etc. prepared for SPIDA 2009+ (development version)
Description
Usage
Arguments
Details
Value
Warning
Author(s)
See Also
Examples
Description
These functions implement a general spline with possibly different degrees in each interval and and different orders of smoothness
at each knot, including the possibility of allowing a discontinuity at a knot. The function sc helps in the construction of linear hypothesis
matrices to estimate and test levels and derivatives of splines at arbitrary points and the saltus
of derivatives that have discontinuities at knots.
Usage
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gsp(x, knots, degree = 3, smooth = pmax(pmin(degree[-1], degree[-length(degree)]) - 1, 0), intercept = 0, signif = 3)
lsp(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
qs(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
cs(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
sc(sp, x, D = 0, type = 1)
smsp(x, knots)
PolyShift( a, n )
The following functions are intended for internal use:
Xf(x, knots, degree = 3, D = 0, right = TRUE, signif = 3)
Xmat(x, degree, D = 0, signif = 3)
Cmat(knots, degree, smooth, intercept = 0, signif = 3)
Arguments
x
value(s) where spline is evaluated
knots
vector of knots
degree
vector giving the degree of the spline in each interval.
Note the number of intervals is equal to the number of knots + 1. A value of 0
corresponds to a constant in the interval. If the spline should evaluate to 0
in the interval, use the intercept argument to specify some value in the
interval at which the spline must evaluate to 0.
smooth
vector with the degree of smoothness at each knot (0 = continuity, 1 = smooth with continuous first derivative, 2 = continuous second derivative, etc. The value -1 allows a discontinuity at the knot.
intercept
value(s) of x at which the spline has value 0, i.e. the value(s) of x for which
yhat is estimated by the intercept term in the model. The default is 0. If NULL,
the spline is not constrained to evaluate to 0 for any x.
periodic
if TRUE generates a period spline on the base interval [0,max(knots)]. A constraint is generated so that the coefficients generate the same values to the right of max(knots) as they do to the right of 0. Note that all knots should be strictly positive.
lin
provides a matrix specifying additional linear contraints on the 'full' parametrization consisting of blocks of polynomials of degree equal to max(degree) in each of the length(knots)+1 intervals of the spline. See below for examples of a spline that is 0 outside of its boundary knots.
signif
number of significant digits used to label coefficients
exclude
number of leading columns to drop from spline matrix: 0: excludes the intercept column,
1: excludes the linear term as well.
Terms that are excluded from the spline matrix can be modeled explicitly.
sp
a spline function defined by gsp. See the examples below.
D
the degree of a derivative: 0: value of the function, 1: first derivative, 2:
second derivative, etc.
type
how a derivative or value of a function is measured at a possible discontinuity
at a knot: 0: limit from the left, 1: limit from the right, 2: saltus (limit from the right minus
the limit from the left)
a
a 'periodic' knot at which a spline repeats itself
n
the maximal 'order', i.e. maximal degree + 1, of a periodic spline
Details
Many polynomial regression splines can be generated by 'plus' functions although the resulting basis for the spline may be ill conditioned. For example a 'quadratic spline' (a spline that is quadratic in each interval with and smooth with a first derivative at each knot) with knots at 1 and 3 can be fitted with:
1
2
All 'standard' polynomial splines with the same degree in each interval and continuity of order one less than the degree at each knot can be constructed in this fashion. A convenient aspect of this parametrization of the spline is that the estimated coefficients have a simple interpretation. The coefficients for 'plus' terms represent the 'saltus' (jump) in the value of a coefficient at the knot. Testing whether the true value of a coefficient is 0 is equivalent to a test for the need to retain the corresponding knot.
This approach does not work for some more complex splines with different degrees or different orders of continuity at the knots. An example is the commonly used natural quadratic spline. A natural quadratic spline with knots at -1,0 and 1 (where -1 and 1 are termed 'boundary knots') is linear in the intervals (-Inf,-1) and (1,+Inf), and quadratic in the intervals (-1,0) and (0,1). The spline is smooth of order 1 at each knot.
Many techniques for fitting splines generate a basis for the spline (columns of the design matrix) that has good numerical properties but whose coefficients do not have a simple interpretation.
The gsp function makes it easy to specify a spline with arbitrary degree in each interval and arbitrary smoothness at each knot. The parametrization produces coefficients that have a simple interpretation. For a spline of degree p at x = 0, coefficients correspond to the 1st, 2nd, ... pth derivative at 0. Additional coefficients correspond to each free saltus at each knot.
The sc function generates a matrix to estimate features of a fitted spline that can be expressed as linear combinations of the spline coefficients. Examples are various derivatives of the spline at any point, left or right derivatives of different orders and the saltus in derivatives at a knot.
A disadvantage of gsp is that the spline basis may be poorly conditioned. The impact of this problem can be mitigated by rescaling the x variable so that it has an appropriate range. For example, with a spline whose highest degree is cubic, ensuring that x has a range between -10 and 10 should avert numerical problems.
gsp generates a matrix of regressors for a spline with knots,
degree of polynomials
in each interval and the degree of smoothness at each knot.
Typically, gsp is used to
define a function that is then used in a model equation. See the examples below.
A function to fit a cubic spline with knots at 5 and 10 is generated with:
1
indicating that a cubic polynomial is used in each of the three
intervals and that the second derivative is continuous at each knot.
A 'natural cubic spline' with linear components in each unbounded interval
would have the form:
1
Quadratic and linear splines, respectively:
1
2
Where the same degree is used for all intervals and knots, it suffices
to give it once:
1
2
An easy way to specify a model in which a knot is dropped is to force
a degree of continuity equal to the degree of adjoining polynomials, e.g.
to drop the knot at 10, use:
1
This is sometimes easier than laboriously rewriting the spline function
for each null hypothesis.
Depending on the maximal degree of the spline, the range of x
should not be excessive to avoid numerical problems. The spline
matrix generated is 'raw' and values of max(abs(x))^max(degree)
may appear in the matrix. For example, for a cubic spline, it
might be desirable to rescale x and/or recenter x so abs(x) < 100
if that is not already the case. Note that the knots need to be
correspondingly transformed.
The naming of coefficients should allow them to be easily interpreted.
For example:
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> zapsmall( gsp ( 0:10, c(3, 7) , c(2,3,2), c(1,1)) )
D1(0) D2(0) C(3).2 C(3).3 C(7).2
f(0) 0 0.0 0.0 0.00000 0.0
f(1) 1 0.5 0.0 0.00000 0.0
f(2) 2 2.0 0.0 0.00000 0.0
f(3) 3 4.5 0.0 0.00000 0.0
f(4) 4 8.0 0.5 0.16667 0.0
f(5) 5 12.5 2.0 1.33333 0.0
f(6) 6 18.0 4.5 4.50000 0.0
f(7) 7 24.5 8.0 10.66667 0.0
f(8) 8 32.0 12.5 20.66667 0.5
f(9) 9 40.5 18.0 34.66667 2.0
f(10) 10 50.0 24.5 52.66667 4.5
The coefficient for the first regressor is the first derivative
at x = 0; for the second regressor, the second derivative at 0;
the third, the saltus (change) in the second derivative at x = 3,
the fourth, the saltus in the third derivative at x = 3 and, finally,
the saltus in the second derivative at x = 7.
Example:
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> sp <- function(x) gsp ( x, c(3, 7) , c(2,3,2), c(1,1))
> zd <- data.frame( x = seq(0,10, .5), y = seq(0,10,.5)^2 + rnorm( 21))
> fit <- lm( y ~ sp( x ), zd)
> summary(fit)
> Ls <-cbind( 0, sc( sp, c(1,2,3,3,3,5,7,7,7,8), D=3,
+ type = c(0,0,0,1,2,0,0,1,2,0)))
> zapsmall( Ls )
D1(0) D2(0) C(3).2 C(3).3 C(7).2
D3(1) 0 0 0 0 0 0
D3(2) 0 0 0 0 0 0
D3(3-) 0 0 0 0 0 0
D3(3+) 0 0 0 0 1 0
D3(3+)-D3(3-) 0 0 0 0 1 0
D3(5) 0 0 0 0 1 0
D3(7-) 0 0 0 0 1 0
D3(7+) 0 0 0 0 0 0
D3(7+)-D3(7-) 0 0 0 0 -1 0
D3(8) 0 0 0 0 0 0
> wald( fit, list( 'third derivatives' = Ls))
numDF denDF F.value p.value
third derivatives 1 15 2.013582 0.17634
Estimate Std.Error DF t-value p-value Lower 0.95 Upper 0.95
D3(1) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(2) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(3-) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(3+) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(3+)-D3(3-) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(5) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(7-) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(7+) 0.000000 0.000000 Inf NaN NaN 0.000000 0.000000
D3(7+)-D3(7-) -0.927625 0.653714 15 -1.419008 0.17634 -2.320984 0.465734
D3(8) 0.000000 0.000000 Inf NaN NaN 0.000000 0.000000
Warning messages:
1: In min(dfs[x != 0]) : no non-missing arguments to min; returning Inf
2: In min(dfs[x != 0]) : no non-missing arguments to min; returning Inf
Note that some coefficients that are 0 by design may lead to invalid
DRs and t-values.
sc generates a portion of a hypothesis matrix for the coefficients
of a general spline constructed with gsp
With:
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Warning: sc will not work correctly if the function defining the spline
transforms the variable, e.g. sp <- function(x) gsp( x/100, knot=2 )
Example:
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simd <- data.frame( age = rep(1:50, 2), y = sin(2*pi*(1:100)/5)+ rnorm(100),
G = rep( c("male","female"), c(50,50)))
sp <- function(x) gsp( x, knots = c(10,25,40), degree = c(1,2,2,1),
smooth = c(1,1,1))
fit <- lm( y ~ sp(age)*G, simd)
xyplot( predict(fit) ~ age , simd, groups = G,type = "l")
summary(fit) # convenient display
# output:
Call:
lm(formula = y ~ sp(age) * G, data = simd)
Residuals:
Min 1Q Median 3Q Max
-2.5249 -0.7765 -0.0760 0.7882 2.6265
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.733267 0.605086 1.212 0.229
sp(age)D1(0) -0.084219 0.055163 -1.527 0.130
sp(age)C(10).2 0.010984 0.006910 1.590 0.115
sp(age)C(25).2 -0.023034 0.012881 -1.788 0.077 .
Gmale -0.307665 0.855721 -0.360 0.720
sp(age)D1(0):Gmale 0.058384 0.078012 0.748 0.456
sp(age)C(10).2:Gmale -0.010556 0.009773 -1.080 0.283
sp(age)C(25).2:Gmale 0.026410 0.018216 1.450 0.150
---
Residual standard error: 1.224 on 92 degrees of freedom
Multiple R-squared: 0.0814, Adjusted R-squared: 0.0115
F-statistic: 1.165 on 7 and 92 DF, p-value: 0.3308
# end of output
L0 <- list( "hat" =
rbind( "females at age=20" = c( 1, sc(sp,20), 0, 0* sc(sp,20)),
"males at age=20" = c( 1, sc(sp,20), 1, 1* sc(sp,20))),
"male-female" = rbind( "at 20" = c( 0 , 0*sc(sp,20), 1, 1*sc(sp,20))))
wald( fit, L0 )
...
L1 <- list("D(yhat)/D(age)"=
rbind( "female at age = 25" = c(0, sc(sp,25,1), 0, 0*sc(sp,25,1)),
"male at x = 25" = c(0, sc(sp,25,1), 0, 1*sc(sp,25,1))))
wald( fit, L1)
# output:
numDF denDF F.value p.value
D(yhat)/D(age) 2 92 1.057307 0.35157
Estimate Std.Error DF t-value p-value Lower 0.95 Upper 0.95
female at age = 25 0.080544 0.056974 92 1.413694 0.16083 -0.032612 0.193700
male at x = 25 -0.019412 0.056974 92 -0.340712 0.73410 -0.132568 0.093744
A periodic spline function can be generated by forcing the coefficients beyond
a periodic knot to repeat the pattern in a previous inteval. For example a periodic
spline of period 1 can be created as follows:
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A periodic spline with additional knots can be created as shown below. Note that constraint matrix 'lin' expresses constraints for the 'full' polynomial parametrization, i.e. polynomials of degree 3 (thus of order 4 when including the constant term) in each of the 4 intervals. The constraint given in 'lin' forces the coefficients beyond the periodic knot at x = 1, to repeat the polynomial in the interval just to the right of x = 0.
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Overview of utility functions:
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Xmat = function( x, degree, D = 0, signif = 3)
design/estimation matrix for f[D](x) where f(x) is polynomial of degree degree.
Xf = function( x, knots, degree = 3, D = 0, right = TRUE , signif = 3)
uses Xmat to form 'full' matrix with blocks determined by knots intervals
Cmat = function( knots, degree, smooth, intercept = 0, signif = 3)
linear constraints
Emat = function( knots, degree, smooth , intercept = FALSE, signif = 3)
estimates - not necessarily a basis
basis = function( X , coef = FALSE )
selects linear independent subset of columns of X
spline.T = function( knots, degree, smooth, intercept = 0, signif = 3 )
full transformation of Xf to spline basis and constraints
spline.E = function( knots, degree, smooth, intercept = 0, signif = 3 )
transformation for spline basis (first r columns of spline.T)
gsp = function( x , knots, degree = 3 , smooth = pmax(pmin( degree[-1],
degree[ - length(degree)]) - 1,0 ), intercept = 0, signif = 3)
Value
gsp returns a matrix generating a spline. cs, qs and lsp return matrices
generating cubic,
quadratic and linear splines respectively.
smsp, whose code is adapted from function in the package lmeSplines,
generates a smoothing spline to be used in the random effects portion of
a call to lme.
Warning
The variables generated by gsp are designed so the coefficients are interpretable as changes
in derivatives at knots. The resulting matrix is not designed to have optimal numerical properties.
The intermediate matrices generated by gsp will contain x raised to the power equal to
the highest degree in degree. The values of x should be scaled to avoid
excessively large values in the spline matrix and ensuing numerical problems.
Author(s)
Monette, G. georges@yorku.ca
See Also
Examples
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## Fitting a quadratic spline
simd <- data.frame( age = rep(1:50, 2), y = sin(2*pi*(1:100)/5) + rnorm(100),
G = rep( c('male','female'), c(50,50)))
# define a function generating the spline
sp <- function(x) gsp( x, knots = c(10,25,40), degree = c(1,2,2,1),
smooth = c(1,1,1))
fit <- lm( y ~ sp(age)*G, simd)
require(lattice)
xyplot( predict(fit) ~ age , simd, groups = G,type = 'l')
summary(fit)
## Linear hypotheses
L <- list( "Overall test of slopes at 20" = rbind(
"Female slope at age 20" = c( F20 <- cbind( 0 , sc( sp, 20, D = 1), 0 , 0 * sc( sp, 20, D = 1))),
"Male slope at age 20" = c( M20 <- cbind( 0 , sc( sp, 20, D = 1), 0 , 1 * sc( sp, 20, D = 1))),
"Difference" = c(M20 - F20))
)
wald( fit, L)
## Right and left second derivatives at knots and saltus
L <- list( "Second derivatives and saltus for female curve at knot at 25" =
cbind( 0, sc( sp, c(25,25,25), D = 2, type =c(0,1,2)), 0,0,0,0))
L
wald( fit, L )
## Smoothing splines
library(nlme)
data(Spruce)
Spruce$all <- 1
range( Spruce$days)
sp <- function(x) smsp ( x, seq( 150, 675, 5))
spruce.fit1 <- lme(logSize ~ days, data=Spruce,
random=list(all= pdIdent(~sp(days) -1),
plot=~1, Tree=~1))
summary(spruce.fit1)
pred <- expand.grid( days = seq( 160, 670, 10), all = 1)
pred$logSize <- predict( spruce.fit1, newdata = pred, level = 1)
require( lattice )
xyplot( logSize ~ days, pred, type = 'l')
gmonette/spida documentation built on May 17, 2019, 7:25 a.m.
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Description Usage Arguments Details Value Warning Author(s) See Also Examples
Description
These functions implement a general spline with possibly different degrees in each interval and and different orders of smoothness
at each knot, including the possibility of allowing a discontinuity at a knot. The function sc helps in the construction of linear hypothesis
matrices to estimate and test levels and derivatives of splines at arbitrary points and the saltus
of derivatives that have discontinuities at knots.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | gsp(x, knots, degree = 3, smooth = pmax(pmin(degree[-1], degree[-length(degree)]) - 1, 0), intercept = 0, signif = 3)
lsp(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
qs(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
cs(x, knots = quantile(x, pc), exclude = 0, pc = c(0.25, 0.75))
sc(sp, x, D = 0, type = 1)
smsp(x, knots)
PolyShift( a, n )
The following functions are intended for internal use:
Xf(x, knots, degree = 3, D = 0, right = TRUE, signif = 3)
Xmat(x, degree, D = 0, signif = 3)
Cmat(knots, degree, smooth, intercept = 0, signif = 3)
|
Arguments
x |
value(s) where spline is evaluated |
knots |
vector of knots |
degree |
vector giving the degree of the spline in each interval.
Note the number of intervals is equal to the number of knots + 1. A value of 0
corresponds to a constant in the interval. If the spline should evaluate to 0
in the interval, use the |
smooth |
vector with the degree of smoothness at each knot (0 = continuity, 1 = smooth with continuous first derivative, 2 = continuous second derivative, etc. The value -1 allows a discontinuity at the knot. |
intercept |
value(s) of x at which the spline has value 0, i.e. the value(s) of x for which yhat is estimated by the intercept term in the model. The default is 0. If NULL, the spline is not constrained to evaluate to 0 for any x. |
periodic |
if TRUE generates a period spline on the base interval [0,max(knots)]. A constraint is generated so that the coefficients generate the same values to the right of max(knots) as they do to the right of 0. Note that all knots should be strictly positive. |
lin |
provides a matrix specifying additional linear contraints on the 'full' parametrization consisting of blocks of polynomials of degree equal to max(degree) in each of the length(knots)+1 intervals of the spline. See below for examples of a spline that is 0 outside of its boundary knots. |
signif |
number of significant digits used to label coefficients |
exclude |
number of leading columns to drop from spline matrix: 0: excludes the intercept column, 1: excludes the linear term as well. Terms that are excluded from the spline matrix can be modeled explicitly. |
sp |
a spline function defined by |
D |
the degree of a derivative: 0: value of the function, 1: first derivative, 2: second derivative, etc. |
type |
how a derivative or value of a function is measured at a possible discontinuity at a knot: 0: limit from the left, 1: limit from the right, 2: saltus (limit from the right minus the limit from the left) |
a |
a 'periodic' knot at which a spline repeats itself |
n |
the maximal 'order', i.e. maximal degree + 1, of a periodic spline |
Details
Many polynomial regression splines can be generated by 'plus' functions although the resulting basis for the spline may be ill conditioned. For example a 'quadratic spline' (a spline that is quadratic in each interval with and smooth with a first derivative at each knot) with knots at 1 and 3 can be fitted with:
1 2 |
All 'standard' polynomial splines with the same degree in each interval and continuity of order one less than the degree at each knot can be constructed in this fashion. A convenient aspect of this parametrization of the spline is that the estimated coefficients have a simple interpretation. The coefficients for 'plus' terms represent the 'saltus' (jump) in the value of a coefficient at the knot. Testing whether the true value of a coefficient is 0 is equivalent to a test for the need to retain the corresponding knot.
This approach does not work for some more complex splines with different degrees or different orders of continuity at the knots. An example is the commonly used natural quadratic spline. A natural quadratic spline with knots at -1,0 and 1 (where -1 and 1 are termed 'boundary knots') is linear in the intervals (-Inf,-1) and (1,+Inf), and quadratic in the intervals (-1,0) and (0,1). The spline is smooth of order 1 at each knot.
Many techniques for fitting splines generate a basis for the spline (columns of the design matrix) that has good numerical properties but whose coefficients do not have a simple interpretation.
The gsp function makes it easy to specify a spline with arbitrary degree in each interval and arbitrary smoothness at each knot. The parametrization produces coefficients that have a simple interpretation. For a spline of degree p at x = 0, coefficients correspond to the 1st, 2nd, ... pth derivative at 0. Additional coefficients correspond to each free saltus at each knot.
The sc function generates a matrix to estimate features of a fitted spline that can be expressed as linear combinations of the spline coefficients. Examples are various derivatives of the spline at any point, left or right derivatives of different orders and the saltus in derivatives at a knot.
A disadvantage of gsp is that the spline basis may be poorly conditioned. The impact of this problem can be mitigated by rescaling the x variable so that it has an appropriate range. For example, with a spline whose highest degree is cubic, ensuring that x has a range between -10 and 10 should avert numerical problems.
gsp generates a matrix of regressors for a spline with knots,
degree of polynomials
in each interval and the degree of smoothness at each knot.
Typically, gsp is used to
define a function that is then used in a model equation. See the examples below.
A function to fit a cubic spline with knots at 5 and 10 is generated with:
1 |
indicating that a cubic polynomial is used in each of the three intervals and that the second derivative is continuous at each knot.
A 'natural cubic spline' with linear components in each unbounded interval would have the form:
1 |
Quadratic and linear splines, respectively:
1 2 |
Where the same degree is used for all intervals and knots, it suffices to give it once:
1 2 |
An easy way to specify a model in which a knot is dropped is to force a degree of continuity equal to the degree of adjoining polynomials, e.g. to drop the knot at 10, use:
1 |
This is sometimes easier than laboriously rewriting the spline function for each null hypothesis.
Depending on the maximal degree of the spline, the range of x should not be excessive to avoid numerical problems. The spline matrix generated is 'raw' and values of max(abs(x))^max(degree) may appear in the matrix. For example, for a cubic spline, it might be desirable to rescale x and/or recenter x so abs(x) < 100 if that is not already the case. Note that the knots need to be correspondingly transformed.
The naming of coefficients should allow them to be easily interpreted. For example:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | > zapsmall( gsp ( 0:10, c(3, 7) , c(2,3,2), c(1,1)) )
D1(0) D2(0) C(3).2 C(3).3 C(7).2
f(0) 0 0.0 0.0 0.00000 0.0
f(1) 1 0.5 0.0 0.00000 0.0
f(2) 2 2.0 0.0 0.00000 0.0
f(3) 3 4.5 0.0 0.00000 0.0
f(4) 4 8.0 0.5 0.16667 0.0
f(5) 5 12.5 2.0 1.33333 0.0
f(6) 6 18.0 4.5 4.50000 0.0
f(7) 7 24.5 8.0 10.66667 0.0
f(8) 8 32.0 12.5 20.66667 0.5
f(9) 9 40.5 18.0 34.66667 2.0
f(10) 10 50.0 24.5 52.66667 4.5
|
The coefficient for the first regressor is the first derivative at x = 0; for the second regressor, the second derivative at 0; the third, the saltus (change) in the second derivative at x = 3, the fourth, the saltus in the third derivative at x = 3 and, finally, the saltus in the second derivative at x = 7.
Example:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | > sp <- function(x) gsp ( x, c(3, 7) , c(2,3,2), c(1,1))
> zd <- data.frame( x = seq(0,10, .5), y = seq(0,10,.5)^2 + rnorm( 21))
> fit <- lm( y ~ sp( x ), zd)
> summary(fit)
> Ls <-cbind( 0, sc( sp, c(1,2,3,3,3,5,7,7,7,8), D=3,
+ type = c(0,0,0,1,2,0,0,1,2,0)))
> zapsmall( Ls )
D1(0) D2(0) C(3).2 C(3).3 C(7).2
D3(1) 0 0 0 0 0 0
D3(2) 0 0 0 0 0 0
D3(3-) 0 0 0 0 0 0
D3(3+) 0 0 0 0 1 0
D3(3+)-D3(3-) 0 0 0 0 1 0
D3(5) 0 0 0 0 1 0
D3(7-) 0 0 0 0 1 0
D3(7+) 0 0 0 0 0 0
D3(7+)-D3(7-) 0 0 0 0 -1 0
D3(8) 0 0 0 0 0 0
> wald( fit, list( 'third derivatives' = Ls))
numDF denDF F.value p.value
third derivatives 1 15 2.013582 0.17634
Estimate Std.Error DF t-value p-value Lower 0.95 Upper 0.95
D3(1) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(2) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(3-) 0.000000 0.000000 15 -1.123777 0.27877 0.000000 0.000000
D3(3+) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(3+)-D3(3-) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(5) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(7-) 0.927625 0.653714 15 1.419008 0.17634 -0.465734 2.320984
D3(7+) 0.000000 0.000000 Inf NaN NaN 0.000000 0.000000
D3(7+)-D3(7-) -0.927625 0.653714 15 -1.419008 0.17634 -2.320984 0.465734
D3(8) 0.000000 0.000000 Inf NaN NaN 0.000000 0.000000
Warning messages:
1: In min(dfs[x != 0]) : no non-missing arguments to min; returning Inf
2: In min(dfs[x != 0]) : no non-missing arguments to min; returning Inf
|
Note that some coefficients that are 0 by design may lead to invalid DRs and t-values.
sc generates a portion of a hypothesis matrix for the coefficients
of a general spline constructed with gsp
With:
1 2 3 4 5 6 7 |
Warning: sc will not work correctly if the function defining the spline
transforms the variable, e.g. sp <- function(x) gsp( x/100, knot=2 )
Example:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | simd <- data.frame( age = rep(1:50, 2), y = sin(2*pi*(1:100)/5)+ rnorm(100),
G = rep( c("male","female"), c(50,50)))
sp <- function(x) gsp( x, knots = c(10,25,40), degree = c(1,2,2,1),
smooth = c(1,1,1))
fit <- lm( y ~ sp(age)*G, simd)
xyplot( predict(fit) ~ age , simd, groups = G,type = "l")
summary(fit) # convenient display
# output:
Call:
lm(formula = y ~ sp(age) * G, data = simd)
Residuals:
Min 1Q Median 3Q Max
-2.5249 -0.7765 -0.0760 0.7882 2.6265
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.733267 0.605086 1.212 0.229
sp(age)D1(0) -0.084219 0.055163 -1.527 0.130
sp(age)C(10).2 0.010984 0.006910 1.590 0.115
sp(age)C(25).2 -0.023034 0.012881 -1.788 0.077 .
Gmale -0.307665 0.855721 -0.360 0.720
sp(age)D1(0):Gmale 0.058384 0.078012 0.748 0.456
sp(age)C(10).2:Gmale -0.010556 0.009773 -1.080 0.283
sp(age)C(25).2:Gmale 0.026410 0.018216 1.450 0.150
---
Residual standard error: 1.224 on 92 degrees of freedom
Multiple R-squared: 0.0814, Adjusted R-squared: 0.0115
F-statistic: 1.165 on 7 and 92 DF, p-value: 0.3308
# end of output
L0 <- list( "hat" =
rbind( "females at age=20" = c( 1, sc(sp,20), 0, 0* sc(sp,20)),
"males at age=20" = c( 1, sc(sp,20), 1, 1* sc(sp,20))),
"male-female" = rbind( "at 20" = c( 0 , 0*sc(sp,20), 1, 1*sc(sp,20))))
wald( fit, L0 )
...
L1 <- list("D(yhat)/D(age)"=
rbind( "female at age = 25" = c(0, sc(sp,25,1), 0, 0*sc(sp,25,1)),
"male at x = 25" = c(0, sc(sp,25,1), 0, 1*sc(sp,25,1))))
wald( fit, L1)
# output:
numDF denDF F.value p.value
D(yhat)/D(age) 2 92 1.057307 0.35157
Estimate Std.Error DF t-value p-value Lower 0.95 Upper 0.95
female at age = 25 0.080544 0.056974 92 1.413694 0.16083 -0.032612 0.193700
male at x = 25 -0.019412 0.056974 92 -0.340712 0.73410 -0.132568 0.093744
|
A periodic spline function can be generated by forcing the coefficients beyond a periodic knot to repeat the pattern in a previous inteval. For example a periodic spline of period 1 can be created as follows:
1 2 3 4 5 6 7 8 9 |
A periodic spline with additional knots can be created as shown below. Note that constraint matrix 'lin' expresses constraints for the 'full' polynomial parametrization, i.e. polynomials of degree 3 (thus of order 4 when including the constant term) in each of the 4 intervals. The constraint given in 'lin' forces the coefficients beyond the periodic knot at x = 1, to repeat the polynomial in the interval just to the right of x = 0.
1 2 3 4 5 6 7 8 9 |
Overview of utility functions:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | Xmat = function( x, degree, D = 0, signif = 3)
design/estimation matrix for f[D](x) where f(x) is polynomial of degree degree.
Xf = function( x, knots, degree = 3, D = 0, right = TRUE , signif = 3)
uses Xmat to form 'full' matrix with blocks determined by knots intervals
Cmat = function( knots, degree, smooth, intercept = 0, signif = 3)
linear constraints
Emat = function( knots, degree, smooth , intercept = FALSE, signif = 3)
estimates - not necessarily a basis
basis = function( X , coef = FALSE )
selects linear independent subset of columns of X
spline.T = function( knots, degree, smooth, intercept = 0, signif = 3 )
full transformation of Xf to spline basis and constraints
spline.E = function( knots, degree, smooth, intercept = 0, signif = 3 )
transformation for spline basis (first r columns of spline.T)
gsp = function( x , knots, degree = 3 , smooth = pmax(pmin( degree[-1],
degree[ - length(degree)]) - 1,0 ), intercept = 0, signif = 3)
|
Value
gsp returns a matrix generating a spline. cs, qs and lsp return matrices
generating cubic,
quadratic and linear splines respectively.
smsp, whose code is adapted from function in the package lmeSplines,
generates a smoothing spline to be used in the random effects portion of
a call to lme.
Warning
The variables generated by gsp are designed so the coefficients are interpretable as changes
in derivatives at knots. The resulting matrix is not designed to have optimal numerical properties.
The intermediate matrices generated by gsp will contain x raised to the power equal to
the highest degree in degree. The values of x should be scaled to avoid
excessively large values in the spline matrix and ensuing numerical problems.
Author(s)
Monette, G. georges@yorku.ca
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | ## Fitting a quadratic spline
simd <- data.frame( age = rep(1:50, 2), y = sin(2*pi*(1:100)/5) + rnorm(100),
G = rep( c('male','female'), c(50,50)))
# define a function generating the spline
sp <- function(x) gsp( x, knots = c(10,25,40), degree = c(1,2,2,1),
smooth = c(1,1,1))
fit <- lm( y ~ sp(age)*G, simd)
require(lattice)
xyplot( predict(fit) ~ age , simd, groups = G,type = 'l')
summary(fit)
## Linear hypotheses
L <- list( "Overall test of slopes at 20" = rbind(
"Female slope at age 20" = c( F20 <- cbind( 0 , sc( sp, 20, D = 1), 0 , 0 * sc( sp, 20, D = 1))),
"Male slope at age 20" = c( M20 <- cbind( 0 , sc( sp, 20, D = 1), 0 , 1 * sc( sp, 20, D = 1))),
"Difference" = c(M20 - F20))
)
wald( fit, L)
## Right and left second derivatives at knots and saltus
L <- list( "Second derivatives and saltus for female curve at knot at 25" =
cbind( 0, sc( sp, c(25,25,25), D = 2, type =c(0,1,2)), 0,0,0,0))
L
wald( fit, L )
## Smoothing splines
library(nlme)
data(Spruce)
Spruce$all <- 1
range( Spruce$days)
sp <- function(x) smsp ( x, seq( 150, 675, 5))
spruce.fit1 <- lme(logSize ~ days, data=Spruce,
random=list(all= pdIdent(~sp(days) -1),
plot=~1, Tree=~1))
summary(spruce.fit1)
pred <- expand.grid( days = seq( 160, 670, 10), all = 1)
pred$logSize <- predict( spruce.fit1, newdata = pred, level = 1)
require( lattice )
xyplot( logSize ~ days, pred, type = 'l')
|
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