# genieclust.compare_partitions¶

Partition similarity scores

genieclust.compare_partitions.adjusted_fm_score(x, y)

The Fowlkes-Mallows index adjusted for chance

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.adjusted_mi_score(x, y)

Adjusted mutual information score $$(\mathrm{AMI}_\mathrm{sum})$$

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.adjusted_rand_score(x, y)

The Rand index adjusted for chance

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.compare_partitions(C)

Computes a series of partition similarity scores

Parameters
Cndarray

A c_contiguous confusion matrix (contingency table) with $$K$$ rows and $$L$$ columns, where $$K \le L$$.

Returns
scoresdict

A dictionary with the following keys:

'ar'

'r'

'afm'

'fm'

'mi'

Mutual information score

'nmi'

Normalised mutual information $$(\mathrm{NMI}_\mathrm{sum})$$

'ami'

Adjusted mutual information $$(\mathrm{AMI}_\mathrm{sum})$$

'nacc'

Normalised accuracy (purity)

'psi'

Pair sets index

Notes

Let x and y represent two partitions of the same set with $$n$$ elements into $$K$$ and $$L$$, respectively, nonempty and pairwise disjoint subsets, e.g., two clusterings of a dataset with $$n$$ observations represented as label vectors. Moreover, let C be the confusion matrix (with $$K$$ rows and $$L$$ columns, $$K \leq L$$) corresponding to x and y, see also genieclust.compare_partitions.confusion_matrix.

This function implements a few scores that aim to quantify the similarity between x and y. Partition similarity scores can be used as external cluster validity measures — for comparing the outputs of clustering algorithms with reference (ground truth) labels, see, e.g., https://github.com/gagolews/clustering_benchmarks_v1 for a suite of benchmark datasets.

Every index except mi_score (which computes the mutual information score) outputs 1 given two identical partitions. Note that partitions are always defined up to a bijection of the set of possible labels, e.g., (1, 1, 2, 1) and (4, 4, 2, 4) represent the same 2-partition.

rand_score gives the Rand score (the “probability” of agreement between the two partitions) and adjusted_rand_score is its version corrected for chance [1] (especially Eqs. (2) and (4) therein): its expected value is 0.0 for two independent partitions. Due to the adjustment, the resulting index might also be negative for some inputs.

Similarly, fm_score gives the Fowlkes-Mallows (FM) score and adjusted_fm_score is its adjusted-for-chance version [1].

Note that both the (unadjusted) Rand and FM scores are bounded from below by $$1/(K+1)$$ if $$K = L$$, hence their adjusted versions are preferred.

mi_score, adjusted_mi_score and normalized_mi_score are information-theoretic indices based on mutual information, see the definition of $$\mathrm{AMI}_\mathrm{sum}$$ and $$\mathrm{NMI}_\mathrm{sum}$$ in [4].

normalized_accuracy is defined as $$(\mathrm{Accuracy}(C_\sigma)-1/L)/(1-1/L)$$, where $$C_\sigma$$ is a version of the confusion matrix for given x and y, $$K \leq L$$, with columns permuted based on the solution to the Maximal Linear Sum Assignment Problem. $$\mathrm{Accuracy}(C_\sigma)$$ is sometimes referred to as Purity, e.g., in [2].

pair_sets_index gives the Pair Sets Index (PSI) adjusted for chance [3], $$K \leq L$$. Pairing is based on the solution to the Linear Sum Assignment Problem of a transformed version of the confusion matrix.

References

1(1,2)

Hubert L., Arabie P., Comparing Partitions, Journal of Classification 2(1), 1985, 193-218.

2

Rendon E., Abundez I., Arizmendi A., Quiroz E.M., Internal versus external cluster validation indexes, International Journal of Computers and Communications 5(1), 2011, 27-34.

3

Rezaei M., Franti P., Set matching measures for external cluster validity, IEEE Transactions on Knowledge and Data Mining 28(8), 2016, 2173-2186. doi:10.1109/TKDE.2016.2551240.

4

Vinh N.X., Epps J., Bailey J., Information theoretic measures for clusterings comparison: Variants, properties, normalization and correction for chance, Journal of Machine Learning Research 11, 2010, 2837-2854.

Examples

>>> x = np.r_[1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2]
>>> y = np.r_[2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1]
>>> C = genieclust.compare_partitions.confusion_matrix(x, y)
>>> C
array([[ 1, 10],
[ 8,  2]])
>>> {k : round(v, 2) for k, v in
...      genieclust.compare_partitions.compare_partitions(C).items()}
{'ar': 0.49, 'r': 0.74, 'fm': 0.73, 'afm': 0.49, 'mi': 0.29, 'nmi': 0.41, 'ami': 0.39, 'nacc': 0.71, 'psi': 0.65}
>>> {k : round(v, 2) for k, v in
...      genieclust.compare_partitions.compare_partitions2(x,y).items()}
{'ar': 0.49, 'r': 0.74, 'fm': 0.73, 'afm': 0.49, 'mi': 0.29, 'nmi': 0.41, 'ami': 0.39, 'nacc': 0.71, 'psi': 0.65}
0.49

genieclust.compare_partitions.compare_partitions2(x, y)

Computes a series of partition similarity scores

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
scoresdict

See genieclust.compare_partitions.compare_partitions.

Notes

Calls genieclust.compare_partitions.compare_partitions(C), where C = genieclust.compare_partitions.confusion_matrix(x, y).

genieclust.compare_partitions.confusion_matrix(x, y)

Computes the confusion matrix for two label vectors

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths.

Returns
Cndarray

A (dense) confusion matrix (contingency table) with max(x)-min(x)+1 rows and max(y)-min(y)+1 columns.

Examples

>>> x = np.r_[1, 2, 1, 2, 2, 2, 3, 1, 2, 1, 2, 1, 2, 2]
>>> y = np.r_[3, 3, 3, 3, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2]
>>> C = genieclust.compare_partitions.confusion_matrix(x, y)
>>> C
array([[1, 0, 4],
[0, 6, 2],
[0, 0, 1]])

genieclust.compare_partitions.fm_score(x, y)

The original Fowlkes-Mallows index (not adjusted for chance)

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.mi_score(x, y)

Mutual information score

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.normalize_confusion_matrix(C)

Applies pivoting to a given confusion matrix

Parameters
Cndarray

A c_contiguous confusion matrix (contingency table).

Returns
ndarray

A normalised confusion matrix of the same shape as C.

genieclust.compare_partitions.confusion_matrix

Determines the confusion matrix

Notes

This function permutes the columns of C so as to relocate the largest elements in each row onto the main diagonal.

It may come in handy whenever C summarises the results generated by clustering algorithms, where actual label values do not matter (e.g., (1, 2, 0) can be remapped to (0, 2, 1) with no change in meaning).

Examples

>>> x = np.r_[1, 2, 1, 2, 2, 2, 3, 1, 2, 1, 2, 1, 2, 2]
>>> y = np.r_[3, 3, 3, 3, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2]
>>> C = genieclust.compare_partitions.confusion_matrix(x, y)
>>> C
array([[1, 0, 4],
[0, 6, 2],
[0, 0, 1]])
>>> genieclust.compare_partitions.normalize_confusion_matrix(C)
array([[4, 0, 1],
[2, 6, 0],
[1, 0, 0]])

genieclust.compare_partitions.normalized_accuracy()

genieclust.compare_partitions.normalized accuracy(x, y)

Normalised accuracy

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.normalized_confusion_matrix(x, y)

Computes the confusion matrix for two label vectors and applies pivoting

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths.

Returns
Cndarray

A (dense) confusion matrix (contingency table) with max(x)-min(x)+1 rows and max(y)-min(y)+1 columns.

Examples

>>> x = np.r_[1, 2, 1, 2, 2, 2, 3, 1, 2, 1, 2, 1, 2, 2]
>>> y = np.r_[3, 3, 3, 3, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2]
>>> genieclust.compare_partitions.normalized_confusion_matrix(x, y)
array([[4, 0, 1],
[2, 6, 0],
[1, 0, 0]])

genieclust.compare_partitions.normalized_mi_score(x, y)

Normalised mutual information score $$(\mathrm{NMI}_\mathrm{sum})$$

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.pair_sets_index(x, y)

Pair Sets Index (PSI) adjusted for chance

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions

Computes multiple similarity scores based on a confusion matrix

genieclust.compare_partitions.compare_partitions2

Computes multiple similarity scores based on two label vectors

Notes

See genieclust.compare_partitions.compare_partitions for more details.

genieclust.compare_partitions.rand_score(x, y)

The original Rand index not adjusted for chance

Parameters
x, yarray_like

Two vectors of “small” integers of identical lengths, representing two partitions of the same set.

Returns
double

Partition similarity measure.

genieclust.compare_partitions.compare_partitions
genieclust.compare_partitions.compare_partitions2