ExpectationNormalizedMetric

class ExpectationNormalizedMetric(base_cls: str | RankBasedMetric | type[RankBasedMetric] | None = None, **kwargs)

Bases: DerivedRankBasedMetric

An adjustment to create an expectation-normalized metric.

\[M^* = \frac{M}{\mathbb{E}[M]}\]

In terms of the affine transformation from DerivedRankBasedMetric, we obtain the following coefficients:

\[\begin{split}\alpha &= \frac{1}{\mathbb{E}[M]} \\ \beta &= 0\end{split}\]

Warning

This requires a closed-form solution to the expected value

Initialize the derived metric.

Parameters:

• base_cls (ClassVar[type[RankBasedMetric] | None]) – the base class, or a hint thereof. If None, use the class-attribute

• kwargs – additional keyword-based parameters used to instantiate the base metric

Attributes Summary

base_cls	The rank-based metric class that this derived metric extends
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
key	Return the key for use in metric result dictionaries.
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

Methods Summary

__call__(ranks[, num_candidates, weights])	Evaluate the metric.
adjust(base_metric_result, num_candidates[, ...])	Adjust base metric results based on the number of candidates.
expected_value(num_candidates[, ...])	Compute expected metric value.
extra_repr()	Generate the extra repr, cf.
get_coefficients(num_candidates[, weights])	Compute the scaling coefficients.
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

base_cls: ClassVar[type[RankBasedMetric] | None] = None

The rank-based metric class that this derived metric extends

binarize: ClassVar[bool] = False

whether the metric needs binarized scores

closed_expectation: ClassVar[bool] = True

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

key

Return the key for use in metric result dictionaries.

needs_candidates: ClassVar[bool] = True

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] = False

whether the metric supports weights

synonyms: ClassVar[Collection[str]] = ()

synonyms for this metric

Methods Documentation

__call__(ranks: ndarray, num_candidates: ndarray | None = None, weights: ndarray | None = None) → float

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

adjust(base_metric_result: float, num_candidates: ndarray, weights: ndarray | None = None) → float

Adjust base metric results based on the number of candidates.

Parameters:

• base_metric_result (float) – the result of the base metric

• num_candidates (ndarray) – the number of candidates

• weights (ndarray | None) – shape: s the weights for the individual ranking tasks

Returns:

the adjusted metric

Return type:

float

Note

since the adjustment only depends on the number of candidates, but not the ranks of the predictions, this method can also be used to adjust published results without access to the trained models.

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 \(\mathcal{U}\left(1, N_i\right)\), where \(N_i\) 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

get_coefficients(num_candidates: ndarray, weights: ndarray | None = None) → AffineTransformationParameters

Compute the scaling coefficients.

Parameters:

• num_candidates (ndarray) – the number of candidates

• weights (ndarray | None) – the weights for the individual ranking tasks

Returns:

a tuple (scale, offset)

Return type:

AffineTransformationParameters

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 \(\mathcal{U}\left(1, N_i\right)\), where \(N_i\) 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 \(\mathcal{U}\left(1, N_i\right)\), where \(N_i\) 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 \(\mathcal{U}\left(1, N_i\right)\), where \(N_i\) 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().