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Indicators

Method
FrontContainer
Exact Hypervolume
dimension_type hypervolume() const
dimension_type hypervolume(key_type reference_point) const
Monte-Carlo Hypervolume
dimension_type hypervolume(size_t sample_size) const
dimension_type hypervolume(size_t sample_size, const key_type &reference_point) const
Cardinality
double coverage(const front &rhs) const
double coverage_ratio(const front &rhs) const
Convergence Indicators
double gd(const front &reference) const
double igd(const front &reference) const
double igd_plus(const front &reference) const
double hausdorff(const front &reference) const
Standard deviation of Convergence Indicators
double std_gd(const front &reference) const
double std_igd(const front &reference) const
double std_igd_plus(const front &reference) const
First Front Distribution
[[nodiscard]] double uniformity() const
[[nodiscard]] double average_distance() const
[[nodiscard]] double average_nearest_distance(size_t k = 5) const
[[nodiscard]] double average_crowding_distance() const
Point Distribution
double crowding_distance(const_iterator element, key_type worst_point, key_type ideal_point) const
double crowding_distance(const_iterator element) const
double crowding_distance(const key_type &point) const
Conflict / Harmony
dimension_type direct_conflict(const size_t a, const size_t b) const
[[nodiscard]] double maxmin_conflict(const size_t a, const size_t b) const
[[nodiscard]] double conflict(const size_t a, const size_t b) const
Normalized Conflict / Harmony
[[nodiscard]] double normalized_direct_conflict(const size_t a, const size_t b) const
[[nodiscard]] double normalized_maxmin_conflict(const size_t a, const size_t b) const
[[nodiscard]] double normalized_conflict(const size_t a, const size_t b) const
ArchiveContainer
Cardinality
double coverage(const front &rhs) const
double coverage_ratio(const front &rhs) const
Convergence Indicators
double gd(const front &reference) const
double igd(const front &reference) const
double igd_plus(const front &reference) const
double hausdorff(const front &reference) const
Standard deviation of Convergence Indicators
double std_gd(const front &reference) const
double std_igd(const front &reference) const
double std_igd_plus(const front &reference) const

Parameters

  • reference_point - point used as reference for the hypervolume calculation. When not provided, it defaults to the nadir() point.
  • sample_size - number of samples for the hypervolume estimate
  • rhs - front or archive being compared
  • reference - Target front. An estimate of the best front possible for the problem.
  • k - number of nearest elements to consider
  • element - element for which we want the crowding distance (see below)
  • key_type - point for which we want the crowding distance (see below)
  • worst_point, ideal_point - reference extreme points for the crowding distance
  • a, b - dimension indices

Return value

  • (see section Indicators for fronts)

Complexity

  • (see section Indicators for fronts)

Notes

The archive indicators use their first front as reference.

Example

Continuing from the previous example:

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// Hypervolume
std::cout << "Exact hypervolume: " << ar.hypervolume(ar.nadir()) << std::endl;
std::cout << "Hypervolume approximation (10000 samples): " << ar.hypervolume(10000, ar.nadir()) << std::endl;

// Coverage
std::cout << "C-metric: " << ar.coverage(ar2) << std::endl;
std::cout << "Coverage ratio: " << ar.coverage_ratio(ar2) << std::endl;
std::cout << "C-metric: " << ar2.coverage(ar) << std::endl;
std::cout << "Coverage ratio: " << ar2.coverage_ratio(ar) << std::endl;

// Convergence
archive<double, 3, unsigned> ar_star({min, max, min});
for (const auto &[p,v] : ar) {
    ar_star(p[0] - 1.0, p[1] + 1.0, p[2] - 1.0) = v;
}
assert(ar.is_completely_dominated_by(ar_star));

std::cout << "GD: " << ar.gd(ar_star) << std::endl;
std::cout << "STDGD: " << ar.std_gd(ar_star) << std::endl;
std::cout << "IGD: " << ar.igd(ar_star) << std::endl;
std::cout << "STDGD: " << ar.std_igd(ar_star) << std::endl;
std::cout << "Hausdorff: " << ar.hausdorff(ar_star) << std::endl;
std::cout << "IGD+: " << ar.igd_plus(ar_star) << std::endl;
std::cout << "STDIGD+: " << ar.std_igd_plus(ar_star) << std::endl;

// Distribution
std::cout << "Uniformity: " << ar.uniformity() << std::endl;
std::cout << "Average distance: " << ar.average_distance() << std::endl;
std::cout << "Average nearest distance: " << ar.average_nearest_distance(5) << std::endl;
auto near_origin = ar.find_nearest({0.0, 0.0, 0.0});
std::cout << "Crowding distance: " << ar.crowding_distance(near_origin) << std::endl;
std::cout << "Average crowding distance: " << ar.average_crowding_distance() << std::endl;

// Correlation
std::cout << "Direct conflict(0,1): " << ar.direct_conflict(0,1) << std::endl;
std::cout << "Normalized direct conflict(0,1): " << ar.normalized_direct_conflict(0,1) << std::endl;
std::cout << "Maxmin conflict(0,1): " << ar.maxmin_conflict(0,1) << std::endl;
std::cout << "Normalized maxmin conflict(0,1): " << ar.normalized_maxmin_conflict(0,1) << std::endl;
std::cout << "Non-parametric conflict(0,1): " << ar.conflict(0,1) << std::endl;
std::cout << "Normalized conflict(0,1): " << ar.normalized_conflict(0,1) << std::endl;

std::cout << "Direct conflict(1,2): " << ar.direct_conflict(1,2) << std::endl;
std::cout << "Normalized direct conflict(1,2): " << ar.normalized_direct_conflict(1,2) << std::endl;
std::cout << "Maxmin conflict(1,2): " << ar.maxmin_conflict(1,2) << std::endl;
std::cout << "Normalized maxmin conflict(1,2): " << ar.normalized_maxmin_conflict(1,2) << std::endl;
std::cout << "Non-parametric conflict(1,2): " << ar.conflict(1,2) << std::endl;
std::cout << "Normalized conflict(1,2): " << ar.normalized_conflict(1,2) << std::endl;

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# Hypervolume
print('Exact hypervolume:', ar.hypervolume(ar.nadir()))
print('Hypervolume approximation (10000 samples):', ar.hypervolume(10000, ar.nadir()))

# Coverage
print('C-metric:', ar.coverage(ar2))
print('Coverage ratio:', ar.coverage_ratio(ar2))
print('C-metric:', ar2.coverage(ar))
print('Coverage ratio:', ar2.coverage_ratio(ar))

# Convergence
ar_star = pareto.archive(['min', 'max', 'min'])
for [p, v] in ar:
    ar_star[p[0] - 1.0, p[1] + 1.0, p[2] - 1.0] = v

print('GD:', ar.gd(ar_star))
print('STDGD:', ar.std_gd(ar_star))
print('IGD:', ar.igd(ar_star))
print('STDGD:', ar.std_igd(ar_star))
print('Hausdorff:', ar.hausdorff(ar_star))
print('IGD+:', ar.igd_plus(ar_star))
print('STDIGD+:', ar.std_igd_plus(ar_star))

# Distribution
print("Uniformity:", ar.uniformity())
print("Average distance:", ar.average_distance())
print("Average nearest distance:", ar.average_nearest_distance(5))
near_origin = next(ar.find_nearest(pareto.point([0.0, 0.0])))
print("Crowding distance:", ar.crowding_distance(near_origin[0]))
print("Average crowding distance:", ar.average_crowding_distance())

# Correlation
print('Direct conflict(0,1):', ar.direct_conflict(0, 1))
print('Normalized direct conflict(0,1):', ar.normalized_direct_conflict(0, 1))
print('Maxmin conflict(0,1):', ar.maxmin_conflict(0, 1))
print('Normalized maxmin conflict(0,1):', ar.normalized_maxmin_conflict(0, 1))
print('Non-parametric conflict(0,1):', ar.conflict(0, 1))
print('Normalized conflict(0,1):', ar.normalized_conflict(0, 1))

print('Direct conflict(1,2):', ar.direct_conflict(1, 2))
print('Normalized direct conflict(1,2):', ar.normalized_direct_conflict(1, 2))
print('Maxmin conflict(1,2):', ar.maxmin_conflict(1, 2))
print('Normalized maxmin conflict(1,2):', ar.normalized_maxmin_conflict(1, 2))
print('Non-parametric conflict(1,2):', ar.conflict(1, 2))
print('Normalized conflict(1,2):', ar.normalized_conflict(1, 2))

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Exact hypervolume: 55.4029
Hypervolume approximation (10000 samples): 56.0535
C-metric: 0
Coverage ratio: 0
C-metric: 1
Coverage ratio: inf
GD: 1.54786
STDGD: 0.0465649
IGD: 1.52137
STDGD: 0.0472864
Hausdorff: 1.54786
IGD+: 1.48592
STDIGD+: 0.0522492
Uniformity: 0.355785
Average distance: 2.75683
Average nearest distance: 1.45177
Crowding distance: 3.04714
Average crowding distance: 4.04349
Direct conflict(0,1): 34.3539
Normalized direct conflict(0,1): 0.360795
Maxmin conflict(0,1): 7.77615
Normalized maxmin conflict(0,1): 0.388808
Non-parametric conflict(0,1): 184
Normalized conflict(0,1): 0.92
Direct conflict(1,2): 32.0107
Normalized direct conflict(1,2): 0.280515
Maxmin conflict(1,2): 5.85805
Normalized maxmin conflict(1,2): 0.292903
Non-parametric conflict(1,2): 146
Normalized conflict(1,2): 0.73