Linear Splines with Convenient Parametrisations.

lspline: Linear Splines with Convenient Parametrisations

• Examples
  • Setting knot locations manually
  • Knots at n equal-length intervals
  • Knots at quantiles of x
• Installation
• Acknowledgements

Linear splines with convenient parametrisations such that

• coefficients are slopes of consecutive segments
• coefficients capture slope change at consecutive knots

Knot locations can be specified

• manually (lspline())
• at breaks dividing the range of x into q equal-frequency intervals (qlspline())
• at breaks dividing the range of x into n equal-width intervals (elspline())

Examples

Examples of using lspline(), qlspline(), and elspline(). We will use the following artificial data with knots at x=5 and x=10

set.seed(666)
n <- 200
d <- data.frame(
  x = scales::rescale(rchisq(n, 6), c(0, 20))
)
d$interval <- findInterval(d$x, c(5, 10), rightmost.closed = TRUE) + 1
d$slope <- c(2, -3, 0)[d$interval]
d$intercept <- c(0, 25, -5)[d$interval]
d$y <- with(d, intercept + slope * x + rnorm(n, 0, 1))

Plotting y against x:

library(ggplot2)
fig <- ggplot(d, aes(x=x, y=y)) + 
  geom_point(aes(shape=as.character(slope))) +
  scale_shape_discrete(name="Slope") +
  theme_bw()
fig

The slopes of the consecutive segments are 2, -3, and 0.

Setting knot locations manually

We can parametrize the spline with slopes of individual segments (default marginal=FALSE):

library(lspline)
m1 <- lm(y ~ lspline(x, c(5, 10)), data=d)
knitr::kable(broom::tidy(m1))

Or parametrize with coeficients measuring change in slope (with marginal=TRUE):

m2 <- lm(y ~ lspline(x, c(5,10), marginal=TRUE), data=d)
knitr::kable(broom::tidy(m2))

The coefficients are

• lspline(x, c(5, 10), marginal = TRUE)1 - the slope of the first segment
• lspline(x, c(5, 10), marginal = TRUE)2 - the change in slope at knot x = 5; it is changing from 2 to -3, so by -5
• lspline(x, c(5, 10), marginal = TRUE)3 - tha change in slope at knot x = 10; it is changing from -3 to 0, so by 3

The two parametrisations (obviously) give identical predicted values:

all.equal( fitted(m1), fitted(m2) )
## [1] TRUE

graphically

fig +
  geom_smooth(method="lm", formula=formula(m1), se=FALSE) +
  geom_vline(xintercept = c(5, 10), linetype=2)

Knots at n equal-length intervals

Function elspline() sets the knots at points dividing the range of x into n equal length intervals.

m3 <- lm(y ~ elspline(x, 3), data=d)
knitr::kable(broom::tidy(m3))

Graphically

fig +
  geom_smooth(aes(group=1), method="lm", formula=formula(m3), se=FALSE, n=200)

Knots at quantiles of x

Function qlspline() sets the knots at points dividing the range of x into q equal-frequency intervals.

m4 <- lm(y ~ qlspline(x, 4), data=d)
knitr::kable(broom::tidy(m4))

Graphically

fig +
  geom_smooth(method="lm", formula=formula(m4), se=FALSE, n=200)

Installation

Stable version from CRAN or development version from GitHub with

devtools::install_github("mbojan/lspline", build_vignettes=TRUE)

Acknowledgements

Inspired by Stata command mkspline and function ares::lspline from Junger & Ponce de Leon (2011). As such, the implementation follows Greene (2003), chapter 7.5.2.

• Greene, William H. (2003) Econometric analysis. Pearson Education
• Junger & Ponce de Leon (2011) ares: Environment air pollution epidemiology: a library for timeseries analysis. R package version 0.7.2 retrieved from CRAN archives.