Htydjtk

I took the data from here and wanted to play around with multidimensional scaling with this data. The data looks like this:

In particular, I want to plot the cities in a 2D space, and see how much it matches their real locations in a geographic map from just the information about how far they are from each other, without any explicit latitude and longitude information. This is my code:

import pandas as pd

import numpy as np

from sklearn import manifold

import matplotlib.pyplot as plt



data = pd.read_csv("european_city_distances.csv", index_col='Cities')



mds = manifold.MDS(n_components=2, dissimilarity="precomputed", random_state=6)

results = mds.fit(data.values)



cities = data.columns

coords = results.embedding_



fig = plt.figure(figsize=(12,10))



plt.subplots_adjust(bottom = 0.1)

plt.scatter(coords[:, 0], coords[:, 1])



for label, x, y in zip(cities, coords[:, 0], coords[:, 1]):

    plt.annotate(

        label,

        xy = (x, y), 

        xytext = (-20, 20),

        textcoords = 'offset points'

    )

plt.show()

Most of the cities seem to be around the correct general location relative to each other, except a few infractions - Dublin is too far away from London, Istanbul is in the wrong location, etc. However, if I give a different random_state value, it produces a different "map". For example, random_state=1 produces the following map, where many of the cities do not seem to be around the correct general location relative to other cities:

What I don't understand is, dimensionality reduction methods are not supposed to have randomness associated with them, and thus should not give different results for different seeds. But it does here; so what does it mean?

The documentation of the sklearn.manifold.MDS function states that random_state is "the generator used to initialize the centers". So, in particular, I guess what I'm asking is, whatever initialization of the centres we choose, shouldn't all of them lead to one unique result?

I get a much more "accurate" map (to my eyes at least) by giving the following hyperparameter values:

mds = manifold.MDS(n_components=2, dissimilarity="euclidean", n_init=100, max_iter=1000, random_state=1)