# The detection of discontinuities in a regression curve or surface. This function uses a comparison of left and right handed nonparametric regression curves to assess the evidence for the presence of one or more discontinuities in a regression curve or surface. A hypothesis test is carried out, under the assumption that the errors in the data are approximately normally distributed. A graphical indication of the locations where the evidence for a discontinuity is strongest is also available. ```r sm.discontinuity(x, y, h, hd, ...) ``` ## Arguments - `x`: a vector or two-column matrix of covariate values. - `y`: a vector of responses observed at the covariate locations. - `h`: a smoothing parameter to be used in the construction of the nonparametric regression estimates. A normal kernel function is used and `h` is its standard deviation(s). However, if this argument is omitted `h` will be selected by an approximate degrees of freedom criterion, controlled by the `df` parameter. See `sm.options` for details. - `hd`: a smoothing parameter to be used in smoothing the differences of the left and right sided nonparametric regression estimates. A normal kernel function is used and `hd` is its standard deviation(s). However, if this argument is omitted `hd` will be set to `h * sqrt(0.25)`, and `h` reset to `h * sqrt(0.75)`, when `x` is a vector When `x` is a matrix, `hd` will be set to `h * sqrt(0.5)` and `h` will be reset to the same value. - ``...``: other optional parameters are passed to the `sm.options` function, through a mechanism which limits their effect only to this call of the function; those relevant for this function are `add`, `eval.points`, `ngrid`, `se`, `band`, `xlab`, `ylab`, `xlim`, `ylim`, `lty`, `col`; see the documentation of `sm.options` for their description. ## Returns a list containing the following items - **p**: the p-value for the test of the null hypothesis that no discontinuities are present. - **sigma**: the estimated standard deviation of the errors. - **eval.points**: the evaluation points of the nonparametric regression estimates. When `x` is a matrix, `eval.points` is also a matrix whose columns define the evaluation grid of each margin of the evaluation rectangle. - **st.diff**: a vector or matrix of standardised differences between the left and right sided estimators at the evaluation points. - **diffmat**: when `x` is a vector, this contains the locations and standardised differences where the latter are greater than 2.5. - **angle**: when `x` is a matrix, this contains the estimated angles at which the standardised differences were constructed. - **h**: the principal smoothing parameter. - **hd**: the smoothing parameter used for double-smoothing (see the reference below). ## Side Effects a plot on the current graphical device is produced, unless the option `display="none"` is set. ## Details The reference below describes the statistical methods used in the function. There are minor differences in some computational details of the implementation. Currently duplicated rows of `x` cause a difficulty in the two covariate case. Duplicated rows should be removed. ## References Bowman, A.W., Pope, A. and Ismail, B. (2006). Detecting discontinuities in nonparametric regression curves and surfaces. **Statistics & Computing**, 16, 377--390. ## See Also `sm.regression`, `sm.options` ## Examples ```r par(mfrow = c(3, 2)) with(nile, { sm.discontinuity(Year, Volume, hd = 0) sm.discontinuity(Year, Volume) ind <- (Year > 1898) plot(Year, Volume) h <- h.select(Year, Volume) sm.regression(Year[!ind], Volume[!ind], h, add = TRUE) sm.regression(Year[ ind], Volume[ ind], h, add = TRUE) hvec <- 1:15 p <- numeric(0) for (h in hvec) { result <- sm.discontinuity(Year, Volume, h, display = "none", verbose = 0) p <- c(p, result$p) } plot(hvec, p, type = "l", ylim = c(0, max(p)), xlab = "h") lines(range(hvec), c(0.05, 0.05), lty = 2) }) with(trawl, { Position <- cbind(Longitude, Latitude) ind <- (Longitude < 143.8) # Remove a repeated point which causes difficulty with sm.discontinuity ind[54] <- FALSE sm.regression(Position[ind,], Score1[ind], theta = 35, phi = 30) sm.discontinuity(Position[ind,], Score1[ind], col = "blue") }) par(mfrow = c(1, 1)) # The following example takes longer to run. # Alternative values for nside are 32 and 64. # Alternative values of yjump are 1 and 0.5. # nside <- 16 # yjump <- 2 # x1 <- seq(0, 1, length = nside) # x2 <- seq(0, 1, length = nside) # x <- expand.grid(x1, x2) # x <- cbind(x1 = x[, 1], x2 = x[, 2]) # y <- rnorm(nside * nside) # ind <- (sqrt((x[, 1] - 0.5)^2 + (x[, 2] - 0.5)^2) <= 0.25) # y[ind] <- y[ind] + yjump # image(x1, x2, matrix(y, ncol = nside)) # sm.discontinuity(x, y, df = 20, add = TRUE) ```

sm.discontinuity function