# lspline: Linear Splines with Convenient Parametrisations

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))

term | estimate | std.error | statistic | p.value
--- | --- | --- | --- | ---
(Intercept) | 0.1343204 | 0.2148116 | 0.6252941 | 0.5325054
lspline(x, c(5, 10))1 | 1.9435458 | 0.0597698 | 32.5171747 | 0.0000000
lspline(x, c(5, 10))2 | -2.9666750 | 0.0503967 | -58.8664832 | 0.0000000
lspline(x, c(5, 10))3 | -0.0335289 | 0.0518601 | -0.6465255 | 0.5186955

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))

term | estimate | std.error | statistic | p.value
--- | --- | --- | --- | ---
(Intercept) | 0.1343204 | 0.2148116 | 0.6252941 | 0.5325054
lspline(x, c(5, 10), marginal = TRUE)1 | 1.9435458 | 0.0597698 | 32.5171747 | 0.0000000
lspline(x, c(5, 10), marginal = TRUE)2 | -4.9102208 | 0.0975908 | -50.3143597 | 0.0000000
lspline(x, c(5, 10), marginal = TRUE)3 | 2.9331462 | 0.0885445 | 33.1262479 | 0.0000000

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))

term | estimate | std.error | statistic | p.value
--- | --- | --- | --- | ---
(Intercept) | 3.5484817 | 0.4603827 | 7.707678 | 0.00e+00
elspline(x, 3)1 | 0.4652507 | 0.1010200 | 4.605529 | 7.40e-06
elspline(x, 3)2 | -2.4908385 | 0.1167867 | -21.328105 | 0.00e+00
elspline(x, 3)3 | 0.9475630 | 0.2328691 | 4.069080 | 6.84e-05

Graphically

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

## Knots at `q`uantiles 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))

term | estimate | std.error | statistic | p.value
--- | --- | --- | --- | ---
(Intercept) | 0.0782285 | 0.3948061 | 0.198144 | 0.8431388
qlspline(x, 4)1 | 2.0398804 | 0.1802724 | 11.315548 | 0.0000000
qlspline(x, 4)2 | 1.2675186 | 0.1471270 | 8.615132 | 0.0000000
qlspline(x, 4)3 | -4.5846478 | 0.1476810 | -31.044273 | 0.0000000
qlspline(x, 4)4 | -0.4965858 | 0.0572115 | -8.679818 | 0.0000000

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.