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Isometric Mapping (ISOMAP)

Non-linear dimensionality reduction through Isometric Mapping

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import pandas as pd
import seaborn as sns
from sklearn.preprocessing import MinMaxScaler
from sklearn.manifold import Isomap

raw_data = pd.read_csv('data/A_multivariate_study_of_variation_in_two_species_of_rock_crab_of_genus_Leptograpsus.csv')

data = raw_data.rename(columns={
    'sp': 'Species',
    'sex': 'Sex',
    'index': 'Index',
    'FL': 'Frontal Lobe',
    'RW': 'Rear Width',
    'CL': 'Carapace Midline',
    'CW': 'Maximum Width',
    'BD': 'Body Depth'})

data['Species'] = data['Species'].map({'B':'Blue', 'O':'Orange'})
data['Sex'] = data['Sex'].map({'M':'Male', 'F':'Female'})
data['Class'] = data.Species + data.Sex

data_columns = ['Frontal Lobe',
                'Rear Width',
                'Carapace Midline',
                'Maximum Width',
                'Body Depth']

data.head()

# generate a class variable for all 4 classes
data['Class'] = data.Species + data.Sex

print(data['Class'].value_counts())
data.head(5)

	Species	Sex	Index	Frontal Lobe	Rear Width	Carapace Midline	Maximum Width	Body Depth	Class
0	Blue	Male	1	8.1	6.7	16.1	19.0	7.0	BlueMale
1	Blue	Male	2	8.8	7.7	18.1	20.8	7.4	BlueMale
2	Blue	Male	3	9.2	7.8	19.0	22.4	7.7	BlueMale
3	Blue	Male	4	9.6	7.9	20.1	23.1	8.2	BlueMale
4	Blue	Male	5	9.8	8.0	20.3	23.0	8.2	BlueMale

# normalize data columns
data_norm = data.copy()
data_norm[data_columns] = MinMaxScaler().fit_transform(data[data_columns])

data_norm.describe()

	Index	Frontal Lobe	Rear Width	Carapace Midline	Maximum Width	Body Depth
count	200.000000	200.000000	200.000000	200.000000	200.000000	200.000000
mean	25.500000	0.527233	0.455365	0.529043	0.515053	0.511645
std	14.467083	0.219832	0.187835	0.216382	0.209919	0.220953
min	1.000000	0.000000	0.000000	0.000000	0.000000	0.000000
25%	13.000000	0.358491	0.328467	0.382219	0.384000	0.341935
50%	25.500000	0.525157	0.459854	0.528875	0.525333	0.503226
75%	38.000000	0.682390	0.569343	0.684650	0.664000	0.677419
max	50.000000	1.000000	1.000000	1.000000	1.000000	1.000000

2-Dimensional Plot

no_components = 2
k_nearest_neighbors = 10

isomap = Isomap(
    n_components=no_components,
    n_neighbors=k_nearest_neighbors)

data_isomap = isomap.fit_transform(data_norm[data_columns])
print('Reconstruction Error: ', isomap.reconstruction_error())
# Reconstruction Error:  0.009501240251169362

data_norm[['ISOMAP1', 'ISOMAP2']] = data_isomap
data_norm.head(1)

	Species	Sex	Index	Frontal Lobe	Rear Width	Carapace Midline	Maximum Width	Body Depth	Class	MDS1	MDS2	ISOMAP1	ISOMAP2
0	Blue	Male	1	0.056604	0.014599	0.042553	0.050667	0.058065	BlueMale	-0.482199	-0.917839	1.091359	0.00803

fig = plt.figure(figsize=(10, 8))
sns.scatterplot(x='ISOMAP1', y='ISOMAP2', hue='Class', data=data_norm)

3-Dimensional Plot

no_components = 3
k_nearest_neighbors = 10

isomap = Isomap(
    n_components=no_components,
    n_neighbors=k_nearest_neighbors)

data_isomap = isomap.fit_transform(data_norm[data_columns])
print('Reconstruction Error: ', isomap.reconstruction_error())
# Reconstruction Error:  0.007640087707465774

data_norm[['ISOMAP1', 'ISOMAP2', 'ISOMAP3']] = data_isomap
data_norm.head(1)

	Species	Sex	Index	Frontal Lobe	Rear Width	Carapace Midline	Maximum Width	Body Depth	Class	ISOMAP1	ISOMAP2	ISOMAP3
0	Blue	Male	1	0.056604	0.014599	0.042553	0.050667	0.058065	BlueMale	1.091359	0.00803	0.117078

class_colours = {
    'BlueMale': '#0027c4', #blue
    'BlueFemale': '#f18b0a', #orange
    'OrangeMale': '#0af10a', # green
    'OrangeFemale': '#ff1500', #red
}

colours = data_norm['Class'].apply(lambda x: class_colours[x])

x=data_norm.ISOMAP1
y=data_norm.ISOMAP2
z=data_norm.ISOMAP3

fig = plt.figure(figsize=(10,10))
plt.title('ISOMAP Data Analysis')
ax = fig.add_subplot(projection='3d')

ax.scatter(xs=x, ys=y, zs=z, s=50, c=colours)