# vsmooth.spline: Vector Cubic Smoothing Spline In VGAM: Vector Generalized Linear and Additive Models

## Description

Fits a vector cubic smoothing spline.

## Usage

 ```1 2 3 4 5 6 7``` ```vsmooth.spline(x, y, w = NULL, df = rep(5, M), spar = NULL, i.constraint = diag(M), x.constraint = diag(M), constraints = list("(Intercepts)" = i.constraint, x = x.constraint), all.knots = FALSE, var.arg = FALSE, scale.w = TRUE, nk = NULL, control.spar = list()) ```

## Arguments

 `x` A vector, matrix or a list. If a list, the `x` component is used. If a matrix, the first column is used. `x` may also be a complex vector, in which case the real part is used, and the imaginary part is used for the response. In this help file, `n` is the number of unique values of `x`. `y` A vector, matrix or a list. If a list, the `y` component is used. If a matrix, all but the first column is used. In this help file, `M` is the number of columns of `y` if there are no constraints on the functions. `w` The weight matrices or the number of observations. If the weight matrices, then this must be a `n`-row matrix with the elements in matrix-band form (see `iam`). If a vector, then these are the number of observations. By default, `w` is the `M` by `M` identity matrix, denoted by `matrix(1, n, M)`. `df` Numerical vector containing the degrees of freedom for each component function (smooth). If necessary, the vector is recycled to have length equal to the number of component functions to be estimated (`M` if there are no constraints), which equals the number of columns of the `x`-constraint matrix. A value of 2 means a linear fit, and each element of `df` should lie between 2 and `n`. The larger the values of `df` the more wiggly the smooths. `spar` Numerical vector containing the non-negative smoothing parameters for each component function (smooth). If necessary, the vector is recycled to have length equal to the number of component functions to be estimated (`M` if there are no constraints), which equals the number of columns of the `x`-constraint matrix. A value of zero means the smooth goes through the data and hence is wiggly. A value of `Inf` may be assigned, meaning the smooth will be linear. By default, the `NULL` value of `spar` means `df` is used to determine the smoothing parameters. `all.knots` Logical. If `TRUE` then each distinct value of `x` will be a knot. By default, only a subset of the unique values of `x` are used; typically, the number of knots is `O(n^0.25)` for `n` large, but if `n <= 40` then all the unique values of `x` are used. `i.constraint` A `M`-row constraint matrix for the intercepts. It must be of full column rank. By default, the constraint matrix for the intercepts is the `M` by `M` identity matrix, meaning no constraints. `x.constraint` A `M`-row constraint matrix for `x`. It must be of full column rank. By default, the constraint matrix for the intercepts is the `M` by `M` identity matrix, meaning no constraints. `constraints` An alternative to specifying `i.constraint` and `x.constraint`, this is a list with two components corresponding to the intercept and `x` respectively. They must both be a `M`-row constraint matrix with full column rank. `var.arg` Logical: return the pointwise variances of the fit? Currently, this corresponds only to the nonlinear part of the fit, and may be wrong. `scale.w` Logical. By default, the weights `w` are scaled so that the diagonal elements have mean 1. `nk` Number of knots. If used, this argument overrides `all.knots`, and must lie between 6 and `n`+2 inclusive. `control.spar` See `smooth.spline`.

## Details

The algorithm implemented is detailed in Yee (2000). It involves decomposing the component functions into a linear and nonlinear part, and using B-splines. The cost of the computation is `O(n M^3)`.

The argument `spar` contains scaled smoothing parameters.

## Value

An object of class `"vsmooth.spline"` (see `vsmooth.spline-class`).

## WARNING

See `vgam` for information about an important bug.

## Note

This function is quite similar to `smooth.spline` but offers less functionality. For example, cross validation is not implemented here. For `M = 1`, the results will be generally different, mainly due to the different way the knots are selected.

The vector cubic smoothing spline which `s()` represents is computationally demanding for large M. The cost is approximately O(n M^3) where n is the number of unique abscissae.

Yet to be done: return the unscaled smoothing parameters.

Thomas W. Yee

## References

Yee, T. W. (2000). Vector Splines and Other Vector Smoothers. Pages 529–534. In: Bethlehem, J. G. and van der Heijde, P. G. M. Proceedings in Computational Statistics COMPSTAT 2000. Heidelberg: Physica-Verlag.

`vsmooth.spline-class`, `plot.vsmooth.spline`, `predict.vsmooth.spline`, `iam`, `sm.os`, `s`, `smooth.spline`.
 ``` 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``` ```nn <- 20; x <- 2 + 5*(nn:1)/nn x[2:4] <- x[5:7] # Allow duplication y1 <- sin(x) + rnorm(nn, sd = 0.13) y2 <- cos(x) + rnorm(nn, sd = 0.13) y3 <- 1 + sin(x) + rnorm(nn, sd = 0.13) # Run this for constraints y <- cbind(y1, y2, y3) ww <- cbind(rep(3, nn), 4, (1:nn)/nn) (fit <- vsmooth.spline(x, y, w = ww, df = 5)) ## Not run: plot(fit) # The 1st and 3rd functions do not differ by a constant ## End(Not run) mat <- matrix(c(1,0,1, 0,1,0), 3, 2) (fit2 <- vsmooth.spline(x, y, w = ww, df = 5, i.constr = mat, x.constr = mat)) # The 1st and 3rd functions do differ by a constant: mycols <- c("orange", "blue", "orange") ## Not run: plot(fit2, lcol = mycols, pcol = mycols, las = 1) p <- predict(fit, x = model.matrix(fit, type = "lm"), deriv = 0) max(abs(depvar(fit) - with(p, y))) # Should be 0; and fit@y is not good par(mfrow = c(3, 1)) ux <- seq(1, 8, len = 100) for (dd in 1:3) { pp <- predict(fit, x = ux, deriv = dd) ## Not run: with(pp, matplot(x, y, type = "l", main = paste("deriv =", dd), lwd = 2, ylab = "", cex.axis = 1.5, cex.lab = 1.5, cex.main = 1.5)) ## End(Not run) } ```