# principal curves by Hauberg on a sphere This function fits a principal curve by Hauberg on the unit 2-sphere. ```r SPC.Hauberg(data, q = 0.1, T = nrow(data), step.size = 1e-3, maxit = 10, type = "Intrinsic", thres = 1e-2, deletePoints = FALSE, plot.proj = FALSE, kernel = "quartic", col1 = "blue", col2 = "green", col3 = "red") ``` ## Arguments - `data`: matrix or data frame consisting of spatial locations with two columns. Each row represents longitude and latitude (denoted by degrees). - `q`: numeric value of the smoothing parameter. The parameter plays the same role, as the bandwidth does in kernel regression, in the `SPC` function. The value should be a numeric value between 0.01 and 0.5. The default is 0.1. - `T`: the number of points making up the resulting curve. The default is 1000. - `step.size`: step size of the `PrincipalCircle` function. The default is 0.001. The resulting principal circle is used by an initialization of the `SPC`. - `maxit`: maximum number of iterations. The default is 30. - `type`: type of mean on sphere. The default is "Intrinsic" and the alternative is "extrinsic". - `thres`: threshold of the stopping condition. The default is 0.01. - `deletePoints`: logical value. The argument is an option of whether to delete points or not. If `deletePoints` is FALSE, this function leaves the points in curves which do not have adjacent data for each expectation step. As a result, the function usually returns a closed curve, i.e. a curve without endpoints. If `deletePoints` is TRUE, this function deletes the points in curves which do not have adjacent data for each expectation step. As a result, The `SPC` function usually returns an open curve, i.e. a curve with endpoints. The default is FALSE. - `plot.proj`: logical value. If the argument is TRUE, the projection line for each data is plotted. The default is FALSE. - `kernel`: kind of kernel function. The default is quartic kernel and alternatives are indicator or Gaussian. - `col1`: color of data. The default is blue. - `col2`: color of points in the principal curves. The default is green. - `col3`: color of resulting principal curves. The default is red. ## Details This function fits a principal curve proposed by Hauberg on the sphere, and requires to load the 'rgl', 'sphereplot', and 'geosphere' R packages. ## Returns plot and a list consisting of - **prin.curves**: spatial locations (denoted by degrees) of points in the resulting principal curves. - **line**: connecting line bewteen points of `prin.curves`. - **converged**: whether or not the algorithm converged. - **iteration**: the number of iterations of the algorithm. - **recon.error**: sum of squared distances from the data to their projections. - **num.dist.pt**: the number of distinct projections. - **plot**: plotting of the data and principal curves. ## References Hauberg, S. (2016). Principal curves on Riemannian manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 1915-1921. Jang-Hyun Kim, Jongmin Lee and Hee-Seok Oh. (2020). Spherical Principal Curves . Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical Principal Curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. . ## Author(s) Jongmin Lee ## Note This function requires to load 'rgl', 'sphereplot', and 'geosphere' R packages. ## See Also `SPC`, `Proj.Hauberg`. ## Examples ```r library(rgl) library(sphereplot) library(geosphere) #### example 1: earthquake data data(Earthquake) names(Earthquake) earthquake <- cbind(Earthquake$longitude, Earthquake$latitude) SPC.Hauberg(earthquake, q = 0.1) #### example 2: waveform data ## longitude and latitude are expressed in degrees n <- 200 alpha <- 1/3; freq <- 4 # amplitude and frequency of wave sigma <- 2 # noise level lon <- seq(-180, 180, length.out = n) # uniformly sampled longitude lat <- alpha * 180/pi * sin(freq * lon * pi/180) + 10. # latitude vector ## add Gaussian noises on latitude vector lat1 <- lat + sigma * rnorm(length(lon)) wave <- cbind(lon, lat1) ## implement principal curves by Hauberg to the waveform data SPC.Hauberg(wave, q = 0.05) ```

SPC.Hauberg function