sortedness 1.230712.2

Measures of projection quality

sortedness

sortedness is the level of agreement between two points regarding to how they rank all remaining points in a dataset. This is valid even for points from different spaces, enabling the measurement of the quality of data transformation processes, often dimensionality reduction. It is less sensitive to irrelevant distortions, and return values in a more meaningful interval, than Kruskal stress formula I.
This Python library / code provides a reference implementation for the functions presented here (paper unavailable until publication).

Overview

Local variants return a value for each provided point. The global variant returns a single value for all points. Any local variant can be used as a global measure by taking the mean value.

Local variants: sortedness(X, X_), pwsortedness(X, X_), rsortedness(X, X_).

Global variant: global_sortedness(X, X_).

Python installation

from package through pip

# Set up a virtualenv. 
python3 -m venv venv
source venv/bin/activate

# Install from PyPI
pip install -U sortedness

from source

git clone https://github.com/sortedness/sortedness
cd sortedness
poetry install

Examples

Sortedness

import numpy as np
from numpy.random import permutation
from sklearn.decomposition import PCA

from sortedness import sortedness

# Some synthetic data.
mean = (1, 2)
cov = np.eye(2)
rng = np.random.default_rng(seed=0)
original = rng.multivariate_normal(mean, cov, size=12)
projected2 = PCA(n_components=2).fit_transform(original)
projected1 = PCA(n_components=1).fit_transform(original)
np.random.seed(0)
projectedrnd = permutation(original)

# Print `min`, `mean`, and `max` values.
s = sortedness(original, original)
print(min(s), sum(s) / len(s), max(s))
"""
1.0 1.0 1.0
"""

s = sortedness(original, projected2)
print(min(s), sum(s) / len(s), max(s))
"""
1.0 1.0 1.0
"""

s = sortedness(original, projected1)
print(min(s), sum(s) / len(s), max(s))
"""
0.393463224666 0.7565797804351666 0.944810120534
"""

s = sortedness(original, projectedrnd)
print(min(s), sum(s) / len(s), max(s))
"""
-0.648305479567 -0.09539895194975 0.397019507592
"""

# Single point fast calculation.
s = sortedness(original, projectedrnd, 2)
print(s)
"""
0.231079547491
"""

Pairwise sortedness

import numpy as np
from numpy.random import permutation
from sklearn.decomposition import PCA

from sortedness import pwsortedness

# Some synthetic data.
mean = (1, 2)
cov = np.eye(2)
rng = np.random.default_rng(seed=0)
original = rng.multivariate_normal(mean, cov, size=12)
projected2 = PCA(n_components=2).fit_transform(original)
projected1 = PCA(n_components=1).fit_transform(original)
np.random.seed(0)
projectedrnd = permutation(original)

# Print `min`, `mean`, and `max` values.
s = pwsortedness(original, original)
print(min(s), sum(s) / len(s), max(s))
"""
1.0 1.0 1.0
"""

s = pwsortedness(original, projected2)
print(min(s), sum(s) / len(s), max(s))
"""
1.0 1.0 1.0
"""

s = pwsortedness(original, projected1)
print(min(s), sum(s) / len(s), max(s))
"""
0.649315577592 0.7534291438323333 0.834601601062
"""

s = pwsortedness(original, projectedrnd)
print(min(s), sum(s) / len(s), max(s))
"""
-0.168611098044 -0.07988253899799999 0.14442446342
"""

# Single point fast calculation.
s = pwsortedness(original, projectedrnd, 2)
print(s)
"""
0.036119718802
"""

Global pairwise sortedness

import numpy as np
from numpy.random import permutation
from sklearn.decomposition import PCA

from sortedness import global_pwsortedness

# Some synthetic data.
mean = (1, 2)
cov = np.eye(2)
rng = np.random.default_rng(seed=0)
original = rng.multivariate_normal(mean, cov, size=12)
projected2 = PCA(n_components=2).fit_transform(original)
projected1 = PCA(n_components=1).fit_transform(original)
np.random.seed(0)
projectedrnd = permutation(original)

# Print measurement result and p-value.
s = global_pwsortedness(original, original)
print(list(s))
"""
[1.0, 3.6741408919675163e-93]
"""

s = global_pwsortedness(original, projected2)
print(list(s))
"""
[1.0, 3.6741408919675163e-93]
"""

s = global_pwsortedness(original, projected1)
print(list(s))
"""
[0.7715617715617715, 5.240847664048334e-20]
"""

s = global_pwsortedness(original, projectedrnd)
print(list(s))
"""
[-0.06107226107226107, 0.46847188611226276]
"""

** Copyright (c) 2023. Davi Pereira dos Santos and Tacito Neves**

TODO

Future work address handling large datasets: approximate sortedness value, and size-insensitive weighting scheme.

Reference

Please use the following reference to cite this work:

@inproceedings {10.2312:eurova.20231093,
booktitle = {EuroVis Workshop on Visual Analytics (EuroVA)},
editor = {Angelini, Marco and El-Assady, Mennatallah},
title = {{Nonparametric Dimensionality Reduction Quality Assessment based on Sortedness of Unrestricted Neighborhood}},
author = {Pereira-Santos, Davi and Neves, Tácito Trindade Araújo Tiburtino and Carvalho, André C. P. L. F. de and Paulovich, Fernando V.},
year = {2023},
publisher = {The Eurographics Association},
ISSN = {2664-4487},
ISBN = {978-3-03868-222-6},
DOI = {10.2312/eurova.20231093}
}

Grants

This work was supported by Wellcome Leap 1kD Program; São Paulo Research Foundation (FAPESP) - grant 2020/09835-1; Cana- dian Institute for Health Research (CIHR) Canadian Research Chairs (CRC) stipend [award number 1024586]; Canadian Foun- dation for Innovation (CFI) John R. Evans Leaders Fund (JELF) [grant number 38835]; Dalhousie Medical Research Fund (DMRF) COVID-19 Research Grant [grant number 603082]; and the Cana- dian Institute for Health Research (CIHR) Project Grant [award number 177968].