Calculating reconstruction error

Description

This function calculates reconstruction error.

Usage

Cal.recon(data, line)

Arguments

Details

This function calculates reconstruction error from the data to the line. Longitude should range from -180 to 180 and latitude from -90 to 90. This function requires to load 'geosphere' R package.

Value

summation of squared distance from the data to the line on the unit sphere.

Author(s)

Jongmin Lee

Examples

library(geosphere)           # This function needs to load 'geosphere' R packages.
data <- rbind(c(0, 0), c(50, -10), c(100, -70))
line <- rbind(c(30, 30), c(-20, 50), c(50, 80))
Cal.recon(data, line)

Crossproduct of vectors

Description

This function performs the cross product of two three-dimensional vectors.

Usage

Crossprod(vec1, vec2)

Arguments

Details

This function performs the cross product of two three-dimensional vectors.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

Examples

Crossprod(c(1, 1, 1), c(5,6,10))

The number of distinct points.

Description

This function calculates the number of distinct point in the given data.

Usage

Dist.pt(data)

Arguments

Details

This function calculates the number of distinct points in the given data.

Value

a numeric.

Author(s)

Jongmin Lee

Examples

Dist.pt(rbind(c(0, 0), c(0, 1), c(1, 0), c(1, 1), c(0, 0)))

Earthquake

Description

It is an earthquake data from the U.S Geological Survey that collect significant earthquakes (8+ Mb magnitude) around the Pacific Ocean since the year 1900. The data are available from (https://www.usgs.gov). Additionally, note that distribution of the data has the following features: 1) scattered, 2) curvilinear one-dimensional structure on the sphere.

Usage

data(Earthquake)

Format

A data frame consisting of time, latitude, longitude, depth, magnitude, etc.

Examples

data(Earthquake)
names(Earthquake)
# collect spatial locations (longitude/latitude) of data.
earthquake <- cbind(Earthquake$longitude, Earthquake$latitude)  
library(rgl)
library(sphereplot)
library(geosphere)
#### example 1: principal geodesic analysis (PGA)
PGA(earthquake)

#### example 2: principal circle
circle <- PrincipalCircle(earthquake)          # get center and radius of principal circle
PC <- GenerateCircle(circle[1:2], circle[3])   # generate Principal circle
# plot
sphereplot::rgl.sphgrid()                      
sphereplot::rgl.sphpoints(earthquake, radius = 1, col = "blue", size = 12)
sphereplot::rgl.sphpoints(PC, radius = 1, col = "red", size = 9)

#### example 3: spherical principal curves (SPC, SPC.Hauberg)
SPC(earthquake)         # spherical principal curves.
SPC.Hauberg(earthquake) # principal curves by Hauberg on sphere.

#### example 4: local principal geodesics (LPG)
LPG(earthquake, scale = 0.5, nu = 0.2, maxpt = 20)
LPG(earthquake, scale = 0.4, nu = 0.3)  

Exponential map

Description

This function performs the exponential map at (0, 0, 1) on the unit 2-sphere.

Usage

Expmap(vec)

Arguments

Details

This function performs exponential map at (0, 0, 1) on the unit sphere. vec is an element of the tangent plane of the unit sphere at (0, 0, 1), and the result is an element of the unit sphere in three-dimensional Euclidean space.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

References

Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23, 995-1005.

See Also

Logmap.

Examples

Expmap(c(1, 2))

Finding Extrinsic Mean

Description

This function identifies the extrinsic mean of data on the unit 2-sphere.

Usage

ExtrinsicMean(data, weights = rep(1, nrow(data)))

Arguments

Details

This function identifies the extrinsic mean of data.

Value

two-dimensional vector.

Note

In the case of spheres, if data set is not contained in a hemisphere, then it is possible that the extrinsic mean of the data set does not exists; for example, a great circle.

Author(s)

Jongmin Lee

References

Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical Principal Curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>. Jang-Hyun Kim, Jongmin Lee and Hee-Seok Oh. (2020). Spherical Principal Curves <arXiv:2003.02578>.

See Also

IntrinsicMean.

Examples

#### comparison of Intrinsic mean and extrinsic mean.
#### example: noisy circular data set.
library(rgl)
library(sphereplot)
library(geosphere)
n <- 500                  # the number of samples.
x <- 360 * runif(n) - 180
sigma <- 5
y <- 60 + sigma * rnorm(n)
simul.circle <- cbind(x, y)
data <- simul.circle
In.mean <- IntrinsicMean(data)
Ex.mean <- ExtrinsicMean(data)
## plot (color of data is "blue"; that of intrinsic mean is "red" and 
## that of extrinsic mean is "green".)
sphereplot::rgl.sphgrid()    
sphereplot::rgl.sphpoints(data, radius = 1, col = "blue", size = 12)
sphereplot::rgl.sphpoints(In.mean[1], In.mean[2], radius = 1, col = "red", size = 12)
sphereplot::rgl.sphpoints(Ex.mean[1], Ex.mean[2], radius = 1, col = "green", size = 12)

Generating circle on sphere

Description

This function makes a circle on the unit 2-sphere.

Usage

GenerateCircle(center, radius, T = 1000)

Arguments

Details

This function makes a circle on the unit 2-sphere.

Value

matrix consisting of spatial locations with two columns.

Author(s)

Jongmin Lee

See Also

PrincipalCircle.

Examples

library(rgl)
library(sphereplot)
library(geosphere)
circle <- GenerateCircle(c(0, 0), 1)
# plot (It requires to load 'rgl', 'sphereplot', and 'geosphere' R package.)
sphereplot::rgl.sphgrid()
sphereplot::rgl.sphpoints(circle[, 1], circle[, 2], radius = 1, col = "blue", size = 12)

Finding Intrinsic Mean

Description

This function calculates the intrinsic mean of data on sphere.

Usage

IntrinsicMean(data, weights = rep(1, nrow(data)), thres = 1e-5)

Arguments

Details

This function calculates the intrinsic mean of data. The intrinsic mean is found by the gradient descent algorithm, which works well if the data is well-localized. In the case of spheres, if data is contained in a hemisphere, then the algorithm converges.

Value

two-dimensional vector.

Author(s)

Jongmin Lee

References

Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23, 995-1005. Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical Principal Curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43. 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>.

See Also

ExtrinsicMean.

Examples

#### comparison of Intrinsic mean and extrinsic mean.
#### example: circular data set.
library(rgl)
library(sphereplot)
library(geosphere)
n <- 500
x <- 360 * runif(n) - 180
sigma <- 5
y <- 60 + sigma * rnorm(n)
simul.circle <- cbind(x, y)
data <- simul.circle
In.mean <- IntrinsicMean(data)
Ex.mean <- ExtrinsicMean(data)
## plot (color of data is "blue"; that of intrinsic mean is "red" and 
## that of extrinsic mean is "green".)
sphereplot::rgl.sphgrid()    
sphereplot::rgl.sphpoints(data, radius = 1, col = "blue", size = 12)
sphereplot::rgl.sphpoints(In.mean[1], In.mean[2], radius = 1, col = "red", size = 12)
sphereplot::rgl.sphpoints(Ex.mean[1], Ex.mean[2], radius = 1, col = "green", size = 12)

Gaussian kernel function

Description

This function returns the value of a Gaussian kernel function.

Usage

Kernel.Gaussian(vec)

Arguments

Details

This function returns the value of a Gaussian kernel function. The value of kernel represents the similarity from origin. The function returns a vector whose length is same as vec.

Value

vector.

Author(s)

Jongmin Lee

See Also

Kernel.indicator, Kernel.quartic.

Examples

Kernel.Gaussian(c(0, 1/2, 1, 2))

Indicator kernel function

Description

This function returns the value of an indicator kernel function.

Usage

Kernel.indicator(vec)

Arguments

Details

This function returns the value of an indicator kernel function. The value of kernel represents similarity from origin. The function returns a vector whose length is same as vec.

Value

vector.

Author(s)

Jongmin Lee

See Also

Kernel.Gaussian, Kernel.quartic.

Examples

Kernel.indicator(c(0, 1/2, 1, 2))

Quartic kernel function

Description

This function returns the value of a quartic kernel function.

Usage

Kernel.quartic(vec)

Arguments

Details

This function returns the value of quartic kernel function. The value of kernel represents similarity from origin. The function returns a vector whose length is same as vec.

Value

vector.

Author(s)

Jongmin Lee

See Also

Kernel.Gaussian, Kernel.indicator.

Examples

Kernel.quartic(c(0, 1/2, 1, 2))

Local principal geodesics

Description

Locally definded principal geodesic analysis.

Usage

LPG(data, scale = 0.04, tau = scale/3, nu = 0, maxpt = 500,
seed = NULL, kernel = "indicator", thres = 1e-4, 
col1 = "blue", col2 = "green", col3 = "red")

Arguments

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

Author(s)

Jongmin Lee

See Also

PGA, SPC, SPC.Hauberg.

Examples

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)

Logarithm map

Description

This function performs the logarithm map at (0, 0, 1) on the unit sphere.

Usage

Logmap(vec)

Arguments

Details

This function performs the logarithm map at (0, 0, 1) on the unit sphere. Note that, vec is normalized to be contained in the unit sphere.

Value

two-dimensional vector.

Author(s)

Jongmin Lee

References

Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23, 995-1005.

See Also

Expmap.

Examples

Logmap(c(1/sqrt(2), 1/sqrt(2), 0))

Principal geodesic analysis

Description

This function performs principal geodesic analysis.

Usage

PGA(data, col1 = "blue", col2 = "red")

Arguments

Details

This function performs principal geodesic analysis.

Value

plot and a list consisting of

Note

This function requires to load 'sphereplot', 'geosphere' and 'rgl' R package.

Author(s)

Jongmin Lee

References

Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23, 995-1005.

See Also

LPG.

Examples

library(rgl)
library(sphereplot)
library(geosphere)
#### example 1: noisy half-great circle data
circle <- GenerateCircle(c(150, 60), radius = pi/2)
half.circle <- circle[circle[, 1] < 0, , drop = FALSE]
sigma <- 2
half.circle <- half.circle + sigma * rnorm(nrow(half.circle))
PGA(half.circle)

#### example 2: noisy S-shaped data 
#### The data consists of two parts: x ~ Uniform[0, 20], y = sqrt(20 * x - x^2) + N(0, sigma^2), 
#### x ~ Uniform[-20, 0], y = -sqrt(-20 * x - x^2) + N(0, sigma^2).
n <- 500                                           
x <- 60 * runif(n)
sigma <- 2                                         
y <- (60 * x - x^2)^(1/2) + sigma * rnorm(n)
simul.S1 <- cbind(x, y)
z <- -60 * runif(n)
w <- -(-60 * z - z^2)^(1/2)+ sigma * rnorm(n)
simul.S2 <- cbind(z, w)
simul.S <- rbind(simul.S1, simul.S2)
PGA(simul.S)

Principal circle on a sphere

Description

This function fits a principal circle on sphere via gradient descent algorithm.

Usage

PrincipalCircle(data, step.size = 1e-3, thres = 1e-5, maxit = 10000)

Arguments

Details

This function fits a principal circle on sphere via gradient descent algorithm. The function returns three-dimensional vectors whose components represent longitude and latitude of the center and the radius of the circle in regular order.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

References

Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>.

Jang-Hyun Kim, Jongmin Lee and Hee-Seok Oh (2020), Spherical principal curves <arXiv:2003.02578>.

See Also

GenerateCircle

Examples

library(rgl)
library(sphereplot)
library(geosphere)
#### example 1: half-great circle data
circle <- GenerateCircle(c(150, 60), radius = pi/2)
half.great.circle <- circle[circle[, 1] < 0, , drop = FALSE]
sigma <- 2
half.great.circle <- half.great.circle + sigma * rnorm(nrow(half.great.circle))
## find a principal circle

PC <- PrincipalCircle(half.great.circle)
result <- GenerateCircle(PC[1:2], PC[3])
## plot
rgl.sphgrid()
rgl.sphpoints(half.great.circle, radius = 1, col = "blue", size = 12)
rgl.sphpoints(result, radius = 1, col = "red", size = 6)

#### example 2: circular data
n <- 700                            
x <- seq(-180, 180, length.out = n)
sigma <- 5                          
y <- 45 + sigma * rnorm(n)
simul.circle <- cbind(x, y)
## find a principal circle
PC <- PrincipalCircle(simul.circle)
result <- GenerateCircle(PC[1:2], PC[3])
## plot
sphereplot::rgl.sphgrid()
sphereplot::rgl.sphpoints(simul.circle, radius = 1, col = "blue", size = 12)
sphereplot::rgl.sphpoints(result, radius = 1, col = "red", size = 6)

#### example 3: earthquake data
data(Earthquake)
names(Earthquake)
earthquake <- cbind(Earthquake$longitude, Earthquake$latitude) 
PC <- PrincipalCircle(earthquake)
result <- GenerateCircle(PC[1:2], PC[3])
## plot
sphereplot::rgl.sphgrid(col.long = "black", col.lat = "black")
sphereplot::rgl.sphpoints(earthquake, radius = 1, col = "blue", size = 12)
sphereplot::rgl.sphpoints(result, radius = 1, col = "red", size = 6)

Projecting the nearest point

Description

This function performs the approximated projection for each data.

Usage

Proj.Hauberg(data, line)

Arguments

Details

This function returns the nearest points in line for each point in the data. The function requires to load the 'geosphere' R package.

Value

matrix consisting of spatial locations with two columns.

Author(s)

Jongmin Lee

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 <arXiv:2003.02578>.

Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>.

See Also

SPC.Hauberg

Examples

library(geosphere)
Proj.Hauberg(rbind(c(0, 0), c(10, -20)), rbind(c(50, 10), c(40, 20), c(30, 30)))

Rotating point on a sphere

Description

Rotate a point on the unit 2-sphere.

Usage

Rotate(pt1, pt2)

Arguments

Details

This function rotates pt2 to the extent that pt1 to spherical coordinate (0, 90). The function returns a point as a form of three-dimensional Euclidean coordinate.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

References

https://en.wikipedia.org/wiki/Rodrigues_rotation_formula

See Also

Rotate.inv.

Examples

## If "pt1" is north pole (= (0, 90)), Rotate() function returns Euclidean coordinate of "pt2".
Rotate(c(0, 90), c(10, 10))   # It returns Euclidean coordinate of spatial location (10, 10).
# The Trans.Euclid() function converts spatial coordinate (10, 10) to Euclidean coordinate.
Trans.Euclid(c(10, 10))       

Rotating point on a sphere

Description

Rotate a point on the unit 2-sphere.

Usage

 Rotate.inv(pt1, pt2)

Arguments

Details

This function rotates pt2 to the extent that the spherical coordinate (0, 90) is rotated to pt1. The function is the inverse of the Rotate function, and returns a point as a form of three-dimensional Euclidean coordinate.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

References

https://en.wikipedia.org/wiki/Rodrigues_rotation_formula

See Also

Rotate.

Examples

## If "pt1" is north pole (= (0, 90)), Rotate.inv() returns Euclidean coordinate of "pt2".
# It returns Euclidean coordinate of spatial location (-100, 80).
Rotate.inv(c(0, 90), c(-100, 80)) 
# It converts spatial coordinate (-100, 80) to Euclidean coordinate.
Trans.Euclid(c(-100, 80))                                                   

Spherical principal curves

Description

This function fits a spherical principal curve.

Usage

SPC(data, q = 0.1, T = nrow(data), step.size = 1e-3, maxit = 10, 
type = "Intrinsic", thres = 0.1, deletePoints = FALSE, plot.proj = FALSE, 
kernel = "quartic", col1 = "blue", col2 = "green", col3 = "red")

Arguments

Details

This function fits a spherical principal curves, and requires to load the 'rgl', 'sphereplot', and 'geosphere' R packages.

Value

plot and a list consisting of

Note

This function requires to load 'rgl', 'sphereplot', and 'geosphere' R packages.

Author(s)

Jongmin Lee

References

Jang-Hyun Kim, Jongmin Lee and Hee-Seok Oh. (2020). Spherical Principal Curves <arXiv:2003.02578>.

Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>.

See Also

SPC.Hauberg.

Examples

library(rgl)
library(sphereplot)
library(geosphere)

#### example 2: waveform data
n <- 200
alpha <- 1/3; freq <- 4                           # amplitude and frequency of wave
sigma1 <- 2; sigma2 <- 10                         # noise levels  
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 + sigma1 * rnorm(length(lon)); lat2 <- lat + sigma2 * rnorm(length(lon))
wave1 <- cbind(lon, lat1); wave2 <- cbind(lon, lat2)
## implement SPC to the (noisy) waveform data
SPC(wave1, q = 0.05)
SPC(wave2, q = 0.05)

#### example 1: earthquake data
data(Earthquake)
names(Earthquake)
earthquake <- cbind(Earthquake$longitude, Earthquake$latitude)   
SPC(earthquake, q = 0.1)
## options 1: plot the projection lines (use option of plot.proj = TRUE)
SPC(earthquake, q = 0.1, plot.proj = TRUE)
## option 2: open principal curves (use option of deletePoints = TRUE)
SPC(earthquake, q = 0.04, deletePoints = TRUE)

principal curves by Hauberg on a sphere

Description

This function fits a principal curve by Hauberg on the unit 2-sphere.

Usage

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

Details

This function fits a principal curve proposed by Hauberg on the sphere, and requires to load the 'rgl', 'sphereplot', and 'geosphere' R packages.

Value

plot and a list consisting of

Note

This function requires to load 'rgl', 'sphereplot', and 'geosphere' R packages.

Author(s)

Jongmin Lee

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 <arXiv:2003.02578>.

Jongmin Lee, Jang-Hyun Kim and Hee-Seok Oh. (2021). Spherical Principal Curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2165-2171. <doi.org/10.1109/TPAMI.2020.3025327>.

See Also

SPC, Proj.Hauberg.

Examples

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)

Transforming into Euclidean coordinate

Description

This function converts a spherical coordinate to a Euclidean coordinate.

Usage

Trans.Euclid(vec)

Arguments

Details

This function converts a two-dimensional spherical coordinate to a three-dimensional Euclidean coordinate. Longitude should be range from -180 to 180 and latitude from -90 to 90.

Value

three-dimensional vector.

Author(s)

Jongmin Lee

See Also

Trans.sph.

Examples

Trans.Euclid(c(0, 0))
Trans.Euclid(c(0, 90))
Trans.Euclid(c(90, 0))
Trans.Euclid(c(180, 0))
Trans.Euclid(c(-90, 0))

Transforming into spherical coordinate

Description

This function converts a Euclidean coordinate to a spherical coordinate.

Usage

Trans.sph(vec)

Arguments

Details

This function converts a three-dimensional Euclidean coordinate to a two-dimensional spherical coordinate. If vec is not in the unit sphere, it is divided by its magnitude so that the result lies on the unit sphere.

Value

two-dimensional vector.

Author(s)

Jongmin Lee

See Also

Trans.Euclid.

Examples

Trans.sph(c(1, 0, 0))
Trans.sph(c(0, 1, 0))
Trans.sph(c(0, 0, 1))
Trans.sph(c(-1, 0 , 0))
Trans.sph(c(0, -1, 0))