HarmonicMeanRank

class HarmonicMeanRank[source]

Bases: RankBasedMetric

The harmonic mean rank.

The harmonic mean rank (HMR) computes the weighted harmonic mean over individual ranks \(\{r_i\}_{i=1}^n\), with weights \(\{w_i\}_{i=1}^n\) (defaulting to \(w_i = 1/n\) when not explicitly provided). Letting \(W = \sum_{i=1}^n w_i\) denote the sum of weights, it is given as the reciprocal of the weighted arithmetic mean of reciprocals:

\[HMR = \frac{W}{\sum_{i=1}^{n} \frac{w_i}{r_i}} = \frac{1}{\frac{1}{W} \sum_{i=1}^{n} \frac{w_i}{r_i}}\]

When weights are uniform (\(w_i = 1/n\)), this reduces to the standard harmonic mean \(\frac{n}{\sum_{i=1}^n r_i^{-1}}\).

The harmonic mean is particularly sensitive to small values (good ranks), making it more robust to outliers with large ranks compared to the arithmetic mean. With PyKEEN’s standard 1-based indexing, the harmonic mean rank lies on the interval \([1, \infty)\) where lower is better.

Note

The harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean, assuming positive values. This inequality holds for both weighted and unweighted cases, provided the same weights are used for all three means.

Warning

The expected value and variance of the harmonic mean rank do not have a simple closed-form solution.

Attributes Summary

`binarize`	whether the metric needs binarized scores
`closed_expectation`	whether there is a closed-form solution of the expectation
`closed_variance`	whether there is a closed-form solution of the variance
`increasing`	whether it is increasing, i.e., larger values are better
`key`	Return the key for use in metric result dictionaries.
`name`	The name of the metric
`needs_candidates`	whether the metric requires the number of candidates for each ranking task
`supported_rank_types`	the supported rank types.
`supports_weights`	whether the metric supports weights
`synonyms`	synonyms for this metric
`value_range`	the value range

Methods Summary

`__call__`(ranks[, num_candidates, weights])	Evaluate the metric.
`expected_value`(num_candidates[, ...])	Compute expected metric value.
`extra_repr`()	Generate the extra repr, cf.
`get_description`()	Get the description.
`get_link`()	Get the link from the docdata.
`get_range`()	Get the math notation for the range of this metric.
`get_sampled_values`(num_candidates, num_samples)	Calculate the metric on sampled rank arrays.
`iter_extra_repr`()	Iterate over the components of the `extra_repr()`.
`numeric_expected_value`(**kwargs)	Compute expected metric value by summation.
`numeric_expected_value_with_ci`(**kwargs)	Estimate expected value with confidence intervals.
`numeric_variance`(**kwargs)	Compute variance by summation.
`numeric_variance_with_ci`(**kwargs)	Estimate variance with confidence intervals.
`std`(num_candidates[, num_samples, weights])	Compute the standard deviation.
`variance`(num_candidates[, num_samples, weights])	Compute variance.

Attributes Documentation

binarize: ClassVar[bool] = False: whether the metric needs binarized scores

closed_expectation: ClassVar[bool] = False: whether there is a closed-form solution of the expectation

closed_variance: ClassVar[bool] = False: whether there is a closed-form solution of the variance

increasing: ClassVar[bool] = False: whether it is increasing, i.e., larger values are better

key: Return the key for use in metric result dictionaries.

name: ClassVar[str] = 'Harmonic Mean Rank (HMR)': The name of the metric

needs_candidates: ClassVar[bool] = False: whether the metric requires the number of candidates for each ranking task

supported_rank_types: ClassVar[Collection[Literal['optimistic', 'realistic', 'pessimistic']]] = ('optimistic', 'realistic', 'pessimistic'): the supported rank types. Most of the time equal to all rank types

supports_weights: ClassVar[bool] = True: whether the metric supports weights

synonyms: ClassVar[Collection[str]] = ('hmr',): synonyms for this metric

value_range: ClassVar[ValueRange] = ValueRange(lower=1, lower_inclusive=True, upper=inf, upper_inclusive=False): the value range

Methods Documentation

__call__(ranks: ndarray, num_candidates: ndarray | None = None, weights: ndarray | None = None) → float[source]

Evaluate the metric.

Parameters:

ranks (ndarray) – shape: s the individual ranks
num_candidates (ndarray | None) – shape: s the number of candidates for each individual ranking task
weights (ndarray | None) – shape: s the weights for the individual ranks

Return type:

float

expected_value(num_candidates: ndarray, num_samples: int | None = None, weights: ndarray | None = None, **kwargs) → float

Compute expected metric value.

The expectation is computed under the assumption that each individual rank follows a discrete uniform distribution \(\mathcal{U}\left(1, C_i\right)\), where \(C_i\) denotes the number of candidates for ranking task \(r_i\).

Parameters:

num_candidates (ndarray) – the number of candidates for each individual rank computation
num_samples (int | None) – the number of samples to use for simulation, if no closed form expected value is implemented
weights (ndarray | None) – shape: s the weights for the individual ranking tasks
kwargs – additional keyword-based parameters passed to get_sampled_values(), if no closed form solution is available

Returns:

the expected value of this metric

Raises:

NoClosedFormError – raised if a closed form expectation has not been implemented and no number of samples are given

Return type:

float

Note

Prefers analytical solution, if available, but falls back to numeric estimation via summation, cf. RankBasedMetric.numeric_expected_value().

extra_repr() → str

Generate the extra repr, cf. :meth`torch.nn.Module.extra_repr`.

Returns:: the extra part of the repr()
Return type:: str

classmethod get_description() → str

Get the description.

Return type:: str

classmethod get_link() → str

Get the link from the docdata.

Return type:: str

classmethod get_range() → str

Get the math notation for the range of this metric.

Return type:: str

get_sampled_values(num_candidates: ndarray, num_samples: int, weights: ndarray | None = None, generator: Generator | None = None, memory_intense: bool = True) → ndarray

Calculate the metric on sampled rank arrays.

Parameters:

num_candidates (ndarray) – shape: s the number of candidates for each ranking task
num_samples (int) – the number of samples
weights (ndarray | None) – shape: s the weights for the individual ranking tasks
generator (Generator | None) – a random state for reproducibility
memory_intense (bool) – whether to use a more memory-intense, but more time-efficient variant

Returns:

shape: (num_samples,) the metric evaluated on num_samples sampled rank arrays

Return type:

ndarray

iter_extra_repr() → Iterable[str]

Iterate over the components of the extra_repr().

This method is typically overridden. A common pattern would be

def iter_extra_repr(self) -> Iterable[str]:
    yield from super().iter_extra_repr()
    yield "<key1>=<value1>"
    yield "<key2>=<value2>"

Returns:: an iterable over individual components of the extra_repr()
Return type:: Iterable[str]

numeric_expected_value(**kwargs) → float

Compute expected metric value by summation.

The expectation is computed under the assumption that each individual rank follows a discrete uniform distribution \(\mathcal{U}\left(1, C_i\right)\), where \(C_i\) denotes the number of candidates for ranking task \(r_i\).

Parameters:: kwargs – keyword-based parameters passed to get_sampled_values()
Returns:: The estimated expected value of this metric
Return type:: float

Warning

Depending on the metric, the estimate may not be very accurate and converge slowly, cf. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_discrete.expect.html

numeric_expected_value_with_ci(**kwargs) → ndarray

Estimate expected value with confidence intervals.

Return type:: ndarray

numeric_variance(**kwargs) → float

Compute variance by summation.

The variance is computed under the assumption that each individual rank follows a discrete uniform distribution \(\mathcal{U}\left(1, C_i\right)\), where \(C_i\) denotes the number of candidates for ranking task \(r_i\).

Parameters:: kwargs – keyword-based parameters passed to get_sampled_values()
Returns:: The estimated variance of this metric
Return type:: float

Warning

Depending on the metric, the estimate may not be very accurate and converge slowly, cf. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_discrete.expect.html

numeric_variance_with_ci(**kwargs) → ndarray

Estimate variance with confidence intervals.

Return type:: ndarray

std(num_candidates: ndarray, num_samples: int | None = None, weights: ndarray | None = None, **kwargs) → float

Compute the standard deviation.

Parameters:

num_candidates (ndarray) – the number of candidates for each individual rank computation
num_samples (int | None) – the number of samples to use for simulation, if no closed form expected value is implemented
weights (ndarray | None) – shape: s the weights for the individual ranking tasks
kwargs – additional keyword-based parameters passed to variance(),

Returns:

The standard deviation (i.e. the square root of the variance) of this metric

Return type:

float

For a detailed explanation, cf. RankBasedMetric.variance().

variance(num_candidates: ndarray, num_samples: int | None = None, weights: ndarray | None = None, **kwargs) → float

Compute variance.

The variance is computed under the assumption that each individual rank follows a discrete uniform distribution \(\mathcal{U}\left(1, C_i\right)\), where \(C_i\) denotes the number of candidates for ranking task \(r_i\).

Parameters:

num_candidates (ndarray) – the number of candidates for each individual rank computation
num_samples (int | None) – the number of samples to use for simulation, if no closed form expected value is implemented
weights (ndarray | None) – shape: s the weights for the individual ranking tasks
kwargs – additional keyword-based parameters passed to get_sampled_values(), if no closed form solution is available

Returns:

The variance of this metric

Raises:

NoClosedFormError – Raised if a closed form variance has not been implemented and no number of samples are given

Return type:

float

Note

Prefers analytical solution, if available, but falls back to numeric estimation via summation, cf. RankBasedMetric.numeric_variance().