Natural splines with knot heights as the basis.
Create the design matrix for a natural spline, such that the coefficient of the resulting fit are the values of the function at the knots.
nsk(x, df = NULL, knots = NULL, intercept = FALSE, b = 0.05,
Boundary.knots = quantile(x, c(b, 1 - b), na.rm = TRUE))x |
the predictor variable. Missing values are allowed. |
df |
degrees of freedom. One can supply df rather than knots; ns() then chooses df - 1 - intercept knots at suitably chosen quantiles of x (which will ignore missing values). The default, df = NULL, sets the number of inner knots as length(knots). |
knots |
breakpoints that define the spline. The default is no knots; together with the natural boundary conditions this results in a basis for linear regression on x. Typical values are the mean or median for one knot, quantiles for more knots. See also Boundary.knots. |
intercept |
if TRUE, an intercept is included in the basis; default is FALSE |
b |
default placement of the boundary knots. A value of
|
Boundary.knots |
boundary points at which to impose the natural boundary conditions and anchor the B-spline basis. Beyond these points the function is assumed to be linear. If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Data can extend beyond Boundary.knots |
The nsk function behaves identically to the ns function,
with two exceptions. The primary one is that the returned basis is
such that coefficients correspond to the value of the fitted function
at the knot points. If intercept = FALSE, there will be k-1
coefficients corresponding to the k knots, and they will be the difference
in predicted value between knots 2-k and knot 1.
The primary advantage to the basis is that the coefficients are
directly interpretable. A second is that tests for the linear and
non-linear components are simple contrasts.
The second differnce with ns is one of opinion with respect to
the default position for the boundary knots. The default here is
closer to that found in the rms::rcs function.
A matrix of dimension length(x) * df where either df was supplied or, if knots were supplied, df = length(knots) + 1 + intercept. Attributes are returned that correspond to the arguments to kns, and explicitly give the knots, Boundary.knots etc for use by predict.kns().
A thin flexible metal or wooden strip is called a spline, and is the traditional method for laying out a smooth curve, e.g., for a ship's hull or an airplane wing. Pins are put into a board and the strip is passed through them, each pin is a 'knot'.
A mathematical spline is a piecewise function between each knot. A linear spline will be a set of connected line segments, a quadratic spline is a set of connected local quadratic functions, constrained to have a continuous first derivative, a cubic spline is cubic between each knot, constrained to have continuous first and second derivatives, and etc. Mathematical splines are not an exact representation of natural splines: being a physical object the wood or metal strip will have continuous derivatives of all orders. Cubic splines are commonly used because they are sufficiently smooth to look natural to the human eye.
If the mathematical spline is further constrained to be linear beyond the end knots, this is often called a 'natural spline', due to the fact that a wooden or metal spline will also be linear beyond the last knots. Another name for the same object is a 'restricted cubic spline', since it is achieved in code by adding a further constraint. Given a vector of data points and a set of knots, it is possible to create a basis matrix X with one column per knot, such that ordinary regression of X on y will fit the cubic spline function, hence these are also called 'regression splines'. One label is no better than another.
Given a basis matrix $X$, the matrix Z= XT for any k by k nonsingular matrix T is is also a basis matrix, and will result in identical predicted values, but a new set of coefficients gamma = (T-inverse) beta in place of beta. One can choose the basis function so that X is easy to construct, to make the regression numerically stable, to make tests easier, or based on other considerations. It seems as though every spline function returns a different basis set, which unfortunately makes fits difficult to compare; and this is yet one more, chosen to make the coefficients more interpretable.
# make some dummy data
tdata <- data.frame(x= lung$age, y = 10*log(lung$age-35) + rnorm(228, 0, 2))
fit1 <- lm(y ~ -1 + nsk(x, df=4, intercept=TRUE) , data=tdata)
fit2 <- lm(y ~ nsk(x, df=3), data=tdata)
# the knots (same for both fits)
knots <- unlist(attributes(fit1$model[[2]])[c('Boundary.knots', 'knots')])
knots
unname(coef(fit1)) # predictions at the knot points
unname(coef(fit1)[-1] - coef(fit1)[1]) # differences: yhat[2:4] - yhat[1]
unname(coef(fit2))
## Not run:
plot(y ~ x, data=tdata)
points(sort(knots), coef(fit1), col=2, pch=19)
coef(fit)[1] + c(0, coef(fit)[-1])
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.