LPG: Local principal geodesics
In spherepc: Spherical Principal Curves
Description
Usage
Arguments
Details
Value
Author(s)
See Also
Examples
Description
Locally definded principal geodesic analysis.
Usage
1
2
3
Arguments
data
matrix or data frame consisting of spatial locations with two columns.
Each row represents longitude and latitude (denoted by degrees).
scale
scale parameter for this function. The argument is the degree to which LPG expresses data locally; thus, as scale grows as the result of LPG become similar to that of the PGA function. The default is 0.4.
tau
forwarding or backwarding distance of each step. It is empirically recommended to choose a third of scale, which is the default of this argument.
nu
parameter to alleviate the bias of resulting curves. nu represents the viscosity of the given data and it should be selected in [0, 1). The default is zero. When the nu is close to 1, the curves usually swirl around, similar to the motion of a large viscous fluid. The swirling can be controlled by the argument maxpt.
maxpt
maximum number of points that each curve has. The default is 500.
seed
random seed number.
kernel
kind of kernel function. The default is the indicator kernel and alternatives are quartic or Gaussian.
thres
threshold of the stopping condition for the IntrinsicMean contained in the LPG function. The default is 1e-4.
col1
color of data. The default is blue.
col2
color of points in the resulting principal curves. The default is green.
col3
color of the resulting curves. The default is red.
Details
Locally definded principal geodesic analysis. The result is sensitive to scale and nu, especially scale should be carefully chosen according to the structure of the given data.
Value
plot and a list consisting of
prin.curves
spatial locations (represented by degrees) of points in the resulting curves.
line
connecting lines between points in prin.curves.
num.curves
the number of the resulting curves.
Author(s)
Jongmin Lee
See Also
PGA, SPC, SPC.Hauberg.
Examples
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library(rgl)
library(sphereplot)
library(geosphere)
#### example 1: spiral data
## longitude and latitude are expressed in degrees
set.seed(40)
n <- 900 # the number of samples
sigma1 <- 1; sigma2 <- 2.5; # noise levels
radius <- 73; slope <- pi/16 # radius and slope of spiral
## polar coordinate of (longitude, latitude)
r <- runif(n)^(2/3) * radius; theta <- -slope * r + 3
## transform to (longitude, latitude)
correction <- (0.5 * r/radius + 0.3) # correction of noise level
lon1 <- r * cos(theta) + correction * sigma1 * rnorm(n)
lat1 <- r * sin(theta) + correction * sigma1 * rnorm(n)
lon2 <- r * cos(theta) + correction * sigma2 * rnorm(n)
lat2 <- r * sin(theta) + correction * sigma2 * rnorm(n)
spiral1 <- cbind(lon1, lat1); spiral2 <- cbind(lon2, lat2)
## plot spiral data
rgl.sphgrid(col.lat = 'black', col.long = 'black')
rgl.sphpoints(spiral1, radius = 1, col = 'blue', size = 12)
## implement the LPG to (noisy) spiral data
LPG(spiral1, scale = 0.06, nu = 0.1, seed = 100)
LPG(spiral2, scale = 0.12, nu = 0.1, seed = 100)
#### example 2: zigzag data set
set.seed(10)
n <- 50 # the number of samples is 6 * n = 300
sigma1 <- 2; sigma2 <- 5 # noise levels
x1 <- x2 <- x3 <- x4 <- x5 <- x6 <- runif(n) * 20 - 20
y1 <- x1 + 20 + sigma1 * rnorm(n); y2 <- -x2 + 20 + sigma1 * rnorm(n)
y3 <- x3 + 60 + sigma1 * rnorm(n); y4 <- -x4 - 20 + sigma1 * rnorm(n)
y5 <- x5 - 20 + sigma1 * rnorm(n); y6 <- -x6 - 60 + sigma1 * rnorm(n)
x <- c(x1, x2, x3, x4, x5, x6); y <- c(y1, y2, y3, y4, y5, y6)
simul.zigzag1 <- cbind(x, y)
## plot zigzag data
sphereplot::rgl.sphgrid(col.lat = 'black', col.long = 'black')
sphereplot::rgl.sphpoints(simul.zigzag1, radius = 1, col = 'blue', size = 12)
## implement the LPG to zigzag data
LPG(simul.zigzag1, scale = 0.1, nu = 0.1, maxpt = 45, seed = 50)
## noisy zigzag data
set.seed(10)
z1 <- z2 <- z3 <- z4 <- z5 <- z6 <- runif(n) * 20 - 20
w1 <- z1 + 20 + sigma2 * rnorm(n); w2 <- -z2 + 20 + sigma2 * rnorm(n)
w3 <- z3 + 60 + sigma2 * rnorm(n); w4 <- -z4 - 20 + sigma2 * rnorm(n)
w5 <- z5 - 20 + sigma2 * rnorm(n); w6 <- -z6 - 60 + sigma2 * rnorm(n)
z <- c(z1, z2, z3, z4, z5, z6); w <- c(w1, w2, w3, w4, w5, w6)
simul.zigzag2 <- cbind(z, w)
## implement the LPG to noisy zigzag data
LPG(simul.zigzag2, scale = 0.2, nu = 0.1, maxpt = 18, seed = 20)
#### example 3: Doubly circular data set
set.seed(30)
n <- 200
sigma <- 1
x1 <- 40 * runif(n) - 20
y1 <- (-x1^2 + 400)^(1/2) + 30 + sigma * rnorm(n)
x2 <- 40 * runif(n) - 20
y2 <- -(-x2^2 + 400)^(1/2) + 30 + sigma * rnorm(n)
y3 <- 40 * runif(n) + 10
x3 <- -(-y3^2 + 60 * y3 - 500)^(1/2) + sigma * rnorm(n)
y4 <- 40 * runif(n) + 10
x4 <- (-y4^2 + 60 * y4 - 500)^(1/2) + sigma * rnorm(n)
Dc1 <- cbind(c(x1, x2, x3, x4), c(y1, y2, y3, y4))
z1 <- 40 * runif(n) - 20
w1 <- (400 - z1^2)^(1/2) - 20 + sigma * rnorm(n)
z2 <- 40 * runif(n) - 20
w2 <- -(400 - z2^2)^(1/2) - 20 + sigma * rnorm(n)
w3 <- -40 * runif(n)
z3 <- (-w3^2 - 40 * w3)^(1/2) + sigma * rnorm(n)
w4 <- -40 * runif(n)
z4 <- -(-w4^2 - 40 * w4)^(1/2) + sigma * rnorm(n)
Dc2 <- cbind(c(z1, z2, z3, z4), c(w1, w2, w3, w4))
Dc <- rbind(Dc1, Dc2)
LPG(Dc, scale = 0.15, nu = 0.1, maxpt = 22,)
#### example 4: real earthquake data
data(Earthquake)
names(Earthquake)
earthquake <- cbind(Earthquake$longitude, Earthquake$latitude)
LPG(earthquake, scale = 0.5, nu = 0.2, maxpt = 20)
LPG(earthquake, scale = 0.4, nu = 0.3)
#### example 5: tree data
## tree consists of stem, branches and subbranches
## generate stem
set.seed(10)
n1 <- 200; n2 <- 100; n3 <- 15 # the number of samples in stem, a branch, and a subbranch
sigma1 <- 0.1; sigma2 <- 0.05; sigma3 <- 0.01 # noise levels
noise1 <- sigma1 * rnorm(n1); noise2 <- sigma2 * rnorm(n2); noise3 <- sigma3 * rnorm(n3)
l1 <- 70; l2 <- 20; l3 <- 1 # length of stem, branches, and subbranches
rep1 <- l1 * runif(n1) # repeated part of stem
stem <- cbind(0 + noise1, rep1 - 10)
## generate branch
rep2 <- l2 * runif(n2) # repeated part of branch
branch1 <- cbind(-rep2, rep2 + 10 + noise2); branch2 <- cbind(rep2, rep2 + noise2)
branch3 <- cbind(rep2, rep2 + 20 + noise2); branch4 <- cbind(rep2, rep2 + 40 + noise2)
branch5 <- cbind(-rep2, rep2 + 30 + noise2)
branch <- rbind(branch1, branch2, branch3, branch4, branch5)
## generate subbranches
rep3 <- l3 * runif(n3) # repeated part in subbranches
branches1 <- cbind(rep3 - 10, rep3 + 20 + noise3)
branches2 <- cbind(-rep3 + 10, rep3 + 10 + noise3)
branches3 <- cbind(rep3 - 14, rep3 + 24 + noise3)
branches4 <- cbind(-rep3 + 14, rep3 + 14 + noise3)
branches5 <- cbind(-rep3 - 12, -rep3 + 22 + noise3)
branches6 <- cbind(rep3 + 12, -rep3 + 12 + noise3)
branches7 <- cbind(-rep3 - 16, -rep3 + 26 + noise3)
branches8 <- cbind(rep3 + 16, -rep3 + 16 + noise3)
branches9 <- cbind(rep3 + 10, -rep3 + 50 + noise3)
branches10 <- cbind(-rep3 - 10, -rep3 + 40 + noise3)
branches11 <- cbind(-rep3 + 12, rep3 + 52 + noise3)
branches12 <- cbind(rep3 - 12, rep3 + 42 + noise3)
branches13 <- cbind(rep3 + 14, -rep3 + 54 + noise3)
branches14 <- cbind(-rep3 - 14, -rep3 + 44 + noise3)
branches15 <- cbind(-rep3 + 16, rep3 + 56 + noise3)
branches16 <- cbind(rep3 - 16, rep3 + 46 + noise3)
branches17 <- cbind(-rep3 + 10, rep3 + 30 + noise3)
branches18 <- cbind(-rep3 + 14, rep3 + 34 + noise3)
branches19 <- cbind(rep3 + 16, -rep3 + 36 + noise3)
branches20 <- cbind(rep3 + 12, -rep3 + 32 + noise3)
sub.branches <- rbind(branches1, branches2, branches3, branches4, branches5, branches6,
+ branches7, branches8, branches9, branches10, branches11, branches12, branches13,
+ branches14, branches15, branches16, branches17, branches18, branches19, branches20)
## tree consists of stem, branch and subbranches
tree <- rbind(stem, branch, sub.branches)
## plot tree data
sphereplot::rgl.sphgrid(col.lat = 'black', col.long = 'black')
sphereplot::rgl.sphpoints(tree, radius = 1, col = 'blue', size = 12)
## implement the LPG function to tree data
LPG(tree, scale = 0.03, nu = 0.2, seed = 10)
spherepc documentation built on Oct. 7, 2021, 9:14 a.m.
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Description Usage Arguments Details Value Author(s) See Also Examples
Description
Locally definded principal geodesic analysis.
Usage
1 2 3 |
Arguments
data |
matrix or data frame consisting of spatial locations with two columns. Each row represents longitude and latitude (denoted by degrees). |
scale |
scale parameter for this function. The argument is the degree to which |
tau |
forwarding or backwarding distance of each step. It is empirically recommended to choose a third of |
nu |
parameter to alleviate the bias of resulting curves. |
maxpt |
maximum number of points that each curve has. The default is 500. |
seed |
random seed number. |
kernel |
kind of kernel function. The default is the indicator kernel and alternatives are quartic or Gaussian. |
thres |
threshold of the stopping condition for the |
col1 |
color of data. The default is blue. |
col2 |
color of points in the resulting principal curves. The default is green. |
col3 |
color of the resulting curves. The default is red. |
Details
Locally definded principal geodesic analysis. The result is sensitive to scale and nu, especially scale should be carefully chosen according to the structure of the given data.
Value
plot and a list consisting of
prin.curves |
spatial locations (represented by degrees) of points in the resulting curves. |
line |
connecting lines between points in |
num.curves |
the number of the resulting curves. |
Author(s)
Jongmin Lee
See Also
PGA, SPC, SPC.Hauberg.
Examples
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 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | library(rgl)
library(sphereplot)
library(geosphere)
#### example 1: spiral data
## longitude and latitude are expressed in degrees
set.seed(40)
n <- 900 # the number of samples
sigma1 <- 1; sigma2 <- 2.5; # noise levels
radius <- 73; slope <- pi/16 # radius and slope of spiral
## polar coordinate of (longitude, latitude)
r <- runif(n)^(2/3) * radius; theta <- -slope * r + 3
## transform to (longitude, latitude)
correction <- (0.5 * r/radius + 0.3) # correction of noise level
lon1 <- r * cos(theta) + correction * sigma1 * rnorm(n)
lat1 <- r * sin(theta) + correction * sigma1 * rnorm(n)
lon2 <- r * cos(theta) + correction * sigma2 * rnorm(n)
lat2 <- r * sin(theta) + correction * sigma2 * rnorm(n)
spiral1 <- cbind(lon1, lat1); spiral2 <- cbind(lon2, lat2)
## plot spiral data
rgl.sphgrid(col.lat = 'black', col.long = 'black')
rgl.sphpoints(spiral1, radius = 1, col = 'blue', size = 12)
## implement the LPG to (noisy) spiral data
LPG(spiral1, scale = 0.06, nu = 0.1, seed = 100)
LPG(spiral2, scale = 0.12, nu = 0.1, seed = 100)
#### example 2: zigzag data set
set.seed(10)
n <- 50 # the number of samples is 6 * n = 300
sigma1 <- 2; sigma2 <- 5 # noise levels
x1 <- x2 <- x3 <- x4 <- x5 <- x6 <- runif(n) * 20 - 20
y1 <- x1 + 20 + sigma1 * rnorm(n); y2 <- -x2 + 20 + sigma1 * rnorm(n)
y3 <- x3 + 60 + sigma1 * rnorm(n); y4 <- -x4 - 20 + sigma1 * rnorm(n)
y5 <- x5 - 20 + sigma1 * rnorm(n); y6 <- -x6 - 60 + sigma1 * rnorm(n)
x <- c(x1, x2, x3, x4, x5, x6); y <- c(y1, y2, y3, y4, y5, y6)
simul.zigzag1 <- cbind(x, y)
## plot zigzag data
sphereplot::rgl.sphgrid(col.lat = 'black', col.long = 'black')
sphereplot::rgl.sphpoints(simul.zigzag1, radius = 1, col = 'blue', size = 12)
## implement the LPG to zigzag data
LPG(simul.zigzag1, scale = 0.1, nu = 0.1, maxpt = 45, seed = 50)
## noisy zigzag data
set.seed(10)
z1 <- z2 <- z3 <- z4 <- z5 <- z6 <- runif(n) * 20 - 20
w1 <- z1 + 20 + sigma2 * rnorm(n); w2 <- -z2 + 20 + sigma2 * rnorm(n)
w3 <- z3 + 60 + sigma2 * rnorm(n); w4 <- -z4 - 20 + sigma2 * rnorm(n)
w5 <- z5 - 20 + sigma2 * rnorm(n); w6 <- -z6 - 60 + sigma2 * rnorm(n)
z <- c(z1, z2, z3, z4, z5, z6); w <- c(w1, w2, w3, w4, w5, w6)
simul.zigzag2 <- cbind(z, w)
## implement the LPG to noisy zigzag data
LPG(simul.zigzag2, scale = 0.2, nu = 0.1, maxpt = 18, seed = 20)
#### example 3: Doubly circular data set
set.seed(30)
n <- 200
sigma <- 1
x1 <- 40 * runif(n) - 20
y1 <- (-x1^2 + 400)^(1/2) + 30 + sigma * rnorm(n)
x2 <- 40 * runif(n) - 20
y2 <- -(-x2^2 + 400)^(1/2) + 30 + sigma * rnorm(n)
y3 <- 40 * runif(n) + 10
x3 <- -(-y3^2 + 60 * y3 - 500)^(1/2) + sigma * rnorm(n)
y4 <- 40 * runif(n) + 10
x4 <- (-y4^2 + 60 * y4 - 500)^(1/2) + sigma * rnorm(n)
Dc1 <- cbind(c(x1, x2, x3, x4), c(y1, y2, y3, y4))
z1 <- 40 * runif(n) - 20
w1 <- (400 - z1^2)^(1/2) - 20 + sigma * rnorm(n)
z2 <- 40 * runif(n) - 20
w2 <- -(400 - z2^2)^(1/2) - 20 + sigma * rnorm(n)
w3 <- -40 * runif(n)
z3 <- (-w3^2 - 40 * w3)^(1/2) + sigma * rnorm(n)
w4 <- -40 * runif(n)
z4 <- -(-w4^2 - 40 * w4)^(1/2) + sigma * rnorm(n)
Dc2 <- cbind(c(z1, z2, z3, z4), c(w1, w2, w3, w4))
Dc <- rbind(Dc1, Dc2)
LPG(Dc, scale = 0.15, nu = 0.1, maxpt = 22,)
#### example 4: real earthquake data
data(Earthquake)
names(Earthquake)
earthquake <- cbind(Earthquake$longitude, Earthquake$latitude)
LPG(earthquake, scale = 0.5, nu = 0.2, maxpt = 20)
LPG(earthquake, scale = 0.4, nu = 0.3)
#### example 5: tree data
## tree consists of stem, branches and subbranches
## generate stem
set.seed(10)
n1 <- 200; n2 <- 100; n3 <- 15 # the number of samples in stem, a branch, and a subbranch
sigma1 <- 0.1; sigma2 <- 0.05; sigma3 <- 0.01 # noise levels
noise1 <- sigma1 * rnorm(n1); noise2 <- sigma2 * rnorm(n2); noise3 <- sigma3 * rnorm(n3)
l1 <- 70; l2 <- 20; l3 <- 1 # length of stem, branches, and subbranches
rep1 <- l1 * runif(n1) # repeated part of stem
stem <- cbind(0 + noise1, rep1 - 10)
## generate branch
rep2 <- l2 * runif(n2) # repeated part of branch
branch1 <- cbind(-rep2, rep2 + 10 + noise2); branch2 <- cbind(rep2, rep2 + noise2)
branch3 <- cbind(rep2, rep2 + 20 + noise2); branch4 <- cbind(rep2, rep2 + 40 + noise2)
branch5 <- cbind(-rep2, rep2 + 30 + noise2)
branch <- rbind(branch1, branch2, branch3, branch4, branch5)
## generate subbranches
rep3 <- l3 * runif(n3) # repeated part in subbranches
branches1 <- cbind(rep3 - 10, rep3 + 20 + noise3)
branches2 <- cbind(-rep3 + 10, rep3 + 10 + noise3)
branches3 <- cbind(rep3 - 14, rep3 + 24 + noise3)
branches4 <- cbind(-rep3 + 14, rep3 + 14 + noise3)
branches5 <- cbind(-rep3 - 12, -rep3 + 22 + noise3)
branches6 <- cbind(rep3 + 12, -rep3 + 12 + noise3)
branches7 <- cbind(-rep3 - 16, -rep3 + 26 + noise3)
branches8 <- cbind(rep3 + 16, -rep3 + 16 + noise3)
branches9 <- cbind(rep3 + 10, -rep3 + 50 + noise3)
branches10 <- cbind(-rep3 - 10, -rep3 + 40 + noise3)
branches11 <- cbind(-rep3 + 12, rep3 + 52 + noise3)
branches12 <- cbind(rep3 - 12, rep3 + 42 + noise3)
branches13 <- cbind(rep3 + 14, -rep3 + 54 + noise3)
branches14 <- cbind(-rep3 - 14, -rep3 + 44 + noise3)
branches15 <- cbind(-rep3 + 16, rep3 + 56 + noise3)
branches16 <- cbind(rep3 - 16, rep3 + 46 + noise3)
branches17 <- cbind(-rep3 + 10, rep3 + 30 + noise3)
branches18 <- cbind(-rep3 + 14, rep3 + 34 + noise3)
branches19 <- cbind(rep3 + 16, -rep3 + 36 + noise3)
branches20 <- cbind(rep3 + 12, -rep3 + 32 + noise3)
sub.branches <- rbind(branches1, branches2, branches3, branches4, branches5, branches6,
+ branches7, branches8, branches9, branches10, branches11, branches12, branches13,
+ branches14, branches15, branches16, branches17, branches18, branches19, branches20)
## tree consists of stem, branch and subbranches
tree <- rbind(stem, branch, sub.branches)
## plot tree data
sphereplot::rgl.sphgrid(col.lat = 'black', col.long = 'black')
sphereplot::rgl.sphpoints(tree, radius = 1, col = 'blue', size = 12)
## implement the LPG function to tree data
LPG(tree, scale = 0.03, nu = 0.2, seed = 10)
|
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