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Examples of joint grid discretization

Jiandong Wang, Sajal Kumar, and Joe Song

2025-12-12

History

Updated: 2025-12-12; 2022-01-27; 2022-01-17; 2020-09-13; 2020-08-01

Created: 2020-03-17

Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters

require(GridOnClusters)
#> Loading required package: GridOnClusters
x = rnorm(500)
y = sin(x)+rnorm(500, sd = 0)
z = cos(x)+rnorm(500, sd = 0)
data = cbind(x, y, z)
ks = 2:20

res = discretize.jointly(data, k=ks, cluster_method = "kmeans+silhouette", 
                         grid_method = "Sort+split") 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "DP exact likelihood") 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC", 
  grid_method = "DP approx likelihood 1-way")
plot(res)
Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.

Example 1. Nonlinear curves using kmeans+silhouette and Ball+BIC clustering with a fixed number of clusters.

Example 2. Nonlinear curves and patterns using kmeans+silhouette and Ball+BIC clustering with a range for the number of clusters

x = rnorm(100)
y = log1p(abs(x))
z = ifelse(x >= -0.5 & x <= 0.5, 0, 1) + rnorm(100, 0, 0.1)
data = cbind(x, y, z)
ks = c(2:5)

#res = discretize.jointly(data, k=ks, cluster_method = "Ball+BIC",
#                         grid_method = "Sort+split", min_level = 1)
#plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "Sort+split", min_level = 1) 

plot(res)

#res = discretize.jointly(data, k=ks, cluster_method = "kmeans+silhouette", 
#                         grid_method = "DP exact likelihood", min_level = 1) 
#plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC", 
  grid_method = "DP exact likelihood", min_level = 1) 
plot(res)
Example 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clustersExample 2. Using a range for the number of kmeans+silhouette and Ball+BIC clusters

Example 2. Using a range for the number of kmeans+silhouette and Ball+BIC clusters

Example 3. Using the partition around medoids clustering method

# using a clustering method other than kmeans+silhouette
x = rnorm(100)
y = log1p(abs(x))
z = sin(x)
data = cbind(x, y, z)

# pre-cluster the data using partition around medoids (PAM)
cluster_label = cluster::pam(x=data, diss = FALSE, metric = "euclidean", k = 4)$clustering

res = discretize.jointly(
  data, cluster_label = cluster_label,
  grid_method = "Sort+split", min_level = 1)
res = discretize.jointly(
  data, cluster_label = cluster_label,
  grid_method = "DP exact likelihood", min_level = 1)
plot(res, main="Original data\nPAM clustering", 
     main.table="Discretized data\nPAM & Sort+split")
Example 3. Using the partition around medoids clustering method.Example 3. Using the partition around medoids clustering method.Example 3. Using the partition around medoids clustering method.Example 3. Using the partition around medoids clustering method.Example 3. Using the partition around medoids clustering method.Example 3. Using the partition around medoids clustering method.

Example 3. Using the partition around medoids clustering method.

Example 4 Random patterns using kmeans+silhouette and Ball+BIC clustering with a range for the number of clusters

ks = 2:20
n = 40*10
sd = 60*4

x=rnorm(2*n, sd=sd)
y=rnorm(2*n, sd=sd)
x=c(x,rnorm(2*n, sd=sd/3))
y=c(y,rnorm(2*n, sd=sd/3)+200)

data = cbind(x, y)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "Sort+split", min_level = 1)
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "Sort+split", min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "DP approx likelihood 1-way", min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "DP approx likelihood 1-way", min_level = 1)
plot(res)
Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.

Example 4. Random patterns using kmeans+silhouette and Ball+BIC clustering with a range.

Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range for the number of clusters

n <- 50*8
ks <- 2:20

X.C1 <- matrix(
   c(rnorm(n, 5, sd=2),
     rnorm(n, 0, sd=40)), 
   ncol = 2, byrow = FALSE
)
X.C2 <- matrix(
   c(rnorm(n, 70, sd=1),
     rnorm(n, 0, sd=1)), 
   ncol = 2, byrow = FALSE
)

X.C3 <- matrix(
   c(rnorm(n, 150, sd=30),
     rnorm(n, 0, sd=30)), 
   ncol = 2, byrow = FALSE
)

data = rbind(X.C1, X.C3)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "Sort+split", min_level = 1)
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "Sort+split", min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "kmeans+silhouette", 
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1, cutoff = 1) 
plot(res)
res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC", 
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)
Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.

Example 5. Multi-cluster random patterns using kmeans+silhouette and Ball+BIC clustering with a range.

Example 6. Exclusive or.

n <- 100
ks <- 2:10

label = c(rep(1, n), rep(2,n), rep(3,n), rep(4,n))

X1 = c(rnorm(n, 0, sd=2),
     rnorm(n, 0, sd=2),
     rnorm(n, 10, sd=2),
     rnorm(n, 10, sd=2))

X2 = c(rnorm(n, 10, sd=2),
     rnorm(n, 0, sd=2),
     rnorm(n, 10, sd=2),
     rnorm(n, 0, sd=2))

X3 = c(rnorm(4*n, 20, sd=10))

X4 = c(rnorm(4*n, 3, sd=20))

data = cbind(X1, X2, X3, X4)

#res = discretize.jointly(data, k=ks, cluster_method = "kmeans+silhouette", 
#                         grid_method = "DP approx likelihood", min_level = 1) 
#plot(res)
#
#res = discretize.jointly(data, k=ks, cluster_method = "kmeans+silhouette", 
#                         grid_method = "DP Compressed majority", min_level = 1) 
#plot(res)

#res = discretize.jointly(data, k=ks, cluster_method = "Ball+BIC", 
#                         grid_method = "DP exact likelihood", min_level = 1) 
#plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC", 
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_label = label,
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)

#res = discretize.jointly(data, k=ks, cluster_label = label,
#                         grid_method = "Sort+split", min_level = 1) 
#plot(res)
#
#res = discretize.jointly(data, k=ks, cluster_method = "kmeans+silhouette", 
#                         grid_method = "Sort+split", min_level = 1) 
#plot(res)
Example 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are randomExample 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are random

Example 6. Exclusive or. Dim1 ⊕ Dim2, Dim3 and Dim4 are random

Example 7. Three well separated clusters

n <- 20
ks <- 2:10

label = c(rep(1, n), rep(2,5*n), rep(3, 5*n))

X1 = c(rnorm(n, 0, sd=1), rnorm(5*n, 15, sd=3), rnorm(5*n, 35, sd=3))
X2 = c(rnorm(n, 0, sd=1), rnorm(5*n, 0, sd=3), rnorm(5*n, 0, sd=3))

data = cbind(X1, X2)

res = discretize.jointly(
  data, cluster_label = label,
  grid_method = "DP exact likelihood", min_level = 1) 
plot(res)
res = discretize.jointly(
  data,  cluster_label = label,
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)

res.entropy = discretize.jointly(
  data,  cluster_label = label,
  grid_method = "DP exact likelihood", min_level = 1, entropy = TRUE) 
plot(res.entropy)
Example 7. Three rounds well seperated on x axisExample 7. Three rounds well seperated on x axisExample 7. Three rounds well seperated on x axisExample 7. Three rounds well seperated on x axisExample 7. Three rounds well seperated on x axisExample 7. Three rounds well seperated on x axis

Example 7. Three rounds well seperated on x axis

Example 8. Four spheres with varying centers and radii

n <- 200
ks <- 2:10

label = c(rep(1, n), rep(2,5*n), rep(3,n), rep(4,3*n))

X1 = c(rnorm(n, 0, sd=3), rnorm(5*n, 15, sd=3), rnorm(n, 5, sd=2), rnorm(3*n, 15, sd=4))

X2 = c(rnorm(n, 0, sd=4), rnorm(5*n, 15, sd=3), rnorm(n, 10, sd=2), rnorm(3*n, 0, sd=4))

data = cbind(X1, X2)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "DP exact likelihood", min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks, cluster_method = "Ball+BIC",
  grid_method = "DP exact likelihood", min_level = 1, entropy = TRUE) 
plot(res)
Example 8. Four sphers with different centers and radiusExample 8. Four sphers with different centers and radiusExample 8. Four sphers with different centers and radiusExample 8. Four sphers with different centers and radiusExample 8. Four sphers with different centers and radiusExample 8. Four sphers with different centers and radius

Example 8. Four sphers with different centers and radius

Example 9. A small dense sphere overlapping a large sphere

Here we evaluate whether a method can recognize the two spheres.

n <- 200
ks <- 2:10

label = c(rep(1, n), rep(2,5*n))

X1 = c(rnorm(n, 0, sd=1), rnorm(5*n, 6, sd=3))
X2 = c(rnorm(n, 0, sd=1), rnorm(5*n, 0, sd=3))

data = cbind(X1, X2)

res = discretize.jointly(
  data, k=ks, cluster_label = label,
  grid_method = "DP exact likelihood", min_level = 1) 
plot(res)

res = discretize.jointly(
  data, k=ks,  cluster_label = label,
  grid_method = "DP approx likelihood 1-way", 
  min_level = 1) 
plot(res)

res.entropy = discretize.jointly(
  data, k=ks,  cluster_label = label,
  grid_method = "DP exact likelihood", min_level = 1, entropy = TRUE) 
plot(res.entropy)
Example 9. A small dense sphere overlapping a large sphere.Example 9. A small dense sphere overlapping a large sphere.Example 9. A small dense sphere overlapping a large sphere.Example 9. A small dense sphere overlapping a large sphere.Example 9. A small dense sphere overlapping a large sphere.Example 9. A small dense sphere overlapping a large sphere.

Example 9. A small dense sphere overlapping a large sphere.

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