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Multidimensional Scaling (MDS)

Multidimensional Scaling is a family of statistical methods that focus on creating mappings of items based on distance. It projects your dataset into a space of fewer dimensions while attempting to preserve the distances of your data points in the original space.

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

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
n_init = 15
metric = True
n_stress='auto'

mds = MDS(
    n_components=no_components,
    n_init=n_init, metric=metric,
    normalized_stress=n_stress)

data_mds = mds.fit_transform(data_norm[data_columns])
print('MSE: ', mds.stress_)
# MSE:  3.886582480465905
# the more components you add the smaller
# the mean squared error becomes - meaning
# your model starts to fit better

data_norm[['MDS1', 'MDS2']] = data_mds
data_norm.head(1)

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

fig = plt.figure(figsize=(10, 8))
sns.scatterplot(x='MDS1', y='MDS2', hue='Class', data=data_norm)

3-Dimensional Plot

no_components = 3
n_init = 15
metric = True
n_stress='auto'

mds = MDS(
    n_components=no_components,
    n_init=n_init, metric=metric,
    normalized_stress=n_stress)

data_mds = mds.fit_transform(data_norm[data_columns])
print('MSE: ', mds.stress_)
# MSE:  2.4601741009431457

data_norm[['MDS1', 'MDS2', 'MDS3']] = data_mds
data_norm.head(1)

	Species	Sex	Index	Frontal Lobe	Rear Width	Carapace Midline	Maximum Width	Body Depth	Class	MDS1	MDS2	MDS3
0	Blue	Male	1	0.056604	0.014599	0.042553	0.050667	0.058065	BlueMale	-0.093961	0.80491	0.645809

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.MDS1
y=data_norm.MDS2
z=data_norm.MDS3

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

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