Performance Indicators

Metaheuristics.jl includes performance indicators to assess evolutionary optimization algorithms performance.

Available indicators:

Performance Indicators

Performance Indicators in Julia

Minimization is always assumed here.

Note that in Metaheuristics.jl, minimization is always assumed. Therefore these indicators have been developed for minimization problems.

Metaheuristics.PerformanceIndicators — Module

PerformanceIndicators

This module includes performance indicators to assess evolutionary multi-objective optimization algorithms.

gd Generational Distance.
igd Inverted Generational Distance.
gd_plus Generational Distance plus.
igd_plus Inverted Generational Distance plus.
covering Covering indicator (C-metric).
hypervolume Hypervolume indicator.

Example

julia> import Metaheuristics: PerformanceIndicators, TestProblems

julia> A = [ collect(1:3) collect(1:3) ]
3×2 Array{Int64,2}:
 1  1
 2  2
 3  3

julia> B = A .- 1
3×2 Array{Int64,2}:
 0  0
 1  1
 2  2

julia> PerformanceIndicators.gd(A, B)
0.47140452079103173

julia> f, bounds, front = TestProblems.get_problem(:ZDT1);

julia> front
                          F space
         ┌────────────────────────────────────────┐ 
       1 │⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
   f_2   │⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⠀⠀⠀⠀⠀│ 
       0 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢄⣀│ 
         └────────────────────────────────────────┘ 
         0                                        1
                            f_1

julia> PerformanceIndicators.igd_plus(front, front)
0.0

source

Generational Distance

Generational Distance in Julia

Metaheuristics.PerformanceIndicators.gd — Function

gd(front, true_pareto_front; p = 1)

Returns the Generational Distance.

Parameters

front and true_pareto_front can be:

N×m matrix where N is the number of points and m is the number of objectives.
State
Array{xFgh_indiv} (usually State.population)

source

Generational Distance Plus

Metaheuristics.PerformanceIndicators.gd_plus — Function

gd_plus(front, true_pareto_front; p = 1)

Returns the Generational Distance Plus.

Parameters

front and true_pareto_front can be:

N×m matrix where N is the number of points and m is the number of objectives.
State
Array{xFgh_indiv} (usually State.population)

source

Inverted Generational Distance

Inverted Generational Distance in Julia

Metaheuristics.PerformanceIndicators.igd — Function

igd(front, true_pareto_front; p = 1)

Returns the Inverted Generational Distance.

Parameters

front and true_pareto_front can be:

N×m matrix where N is the number of points and m is the number of objectives.
State
Array{xFgh_indiv} (usually State.population)

source

Inverted Generational Distance Plus

Metaheuristics.PerformanceIndicators.igd_plus — Function

igd_plus(front, true_pareto_front; p = 1)

Returns the Inverted Generational Distance Plus.

Parameters

front and true_pareto_front can be:

N×m matrix where N is the number of points and m is the number of objectives.
State
Array{xFgh_indiv} (usually State.population)

source

Spacing Indicator

Metaheuristics.PerformanceIndicators.spacing — Function

spacing(A)

Computes the Schott spacing indicator. spacing(A) == 0 means that vectors in A are uniformly distributed.

source

Covering Indicator ($C$-metric)

Metaheuristics.PerformanceIndicators.covering — Function

covering(A, B)

Computes the covering indicator (percentage of vectors in B that are dominated by vectors in A) from two sets with non-dominated solutions.

A and B with size (n, m) where n is number of samples and m is the vector dimension.

Note that covering(A, B) == 1 means that all solutions in B are dominated by those in A. Moreover, covering(A, B) != covering(B, A) in general.

If A::State and B::State, then computes covering(A.population, B.population) after ignoring dominated solutions in each set.

source

Hypervolume

Hypervolume Indicator in Julia

Metaheuristics.PerformanceIndicators.hypervolume — Function

hypervolume(front, reference_point)

Computes the hypervolume indicator, i.e., volume between points in front and reference_point.

Note that each point in front must (weakly) dominates to reference_point. Also, front is a non-dominated set.

If front::State and reference_point::Vector, then computes hypervolume(front.population, reference_point) after ignoring solutions in front that do not dominate reference_point.

source

Examples

Computing hypervolume indicator from vectors in a Matrix

julia> import Metaheuristics.PerformanceIndicators: hypervolume
julia> f1 = collect(0:10); # objective 1
julia> f2 = 10 .- collect(0:10); # objective 2
julia> front = [ f1 f2 ]11×2 Array{Int64,2}:
  0  10
  1   9
  2   8
  3   7
  4   6
  5   5
  6   4
  7   3
  8   2
  9   1
 10   0
julia> reference_point = [11, 11]2-element Array{Int64,1}:
 11
 11
julia> hv = hypervolume(front, reference_point)66.0

Now, let's compute the hypervolume implementation in Julia from the result of NSGA3 when solving DTLZ2 test problem.

julia> using Metaheuristics
julia> import Metaheuristics.PerformanceIndicators: hypervolume
julia> import Metaheuristics: TestProblems, get_non_dominated_solutions
julia> f, bounds, true_front = TestProblems.DTLZ2();
julia> result = optimize(f, bounds, NSGA3());
julia> approx_front = get_non_dominated_solutions(result.population)100-element Array{Metaheuristics.xFgh_solution{Array{Float64,1}},1}:
 (f = [0.5826982323549833, 0.7314005580032928, 0.36835523009765336], g = [0.0], h = [0.0], x = [2.389e-01, 5.717e-01, …, 5.161e-01])
 (f = [0.484082007302127, 0.33926718392046085, 0.8111502065936869], g = [0.0], h = [0.0], x = [5.991e-01, 3.892e-01, …, 5.152e-01])
 (f = [0.429343998023314, 0.6662489712336719, 0.6231893653107466], g = [0.0], h = [0.0], x = [4.242e-01, 6.356e-01, …, 4.674e-01])
 (f = [0.3542572096813144, 0.9322831854054733, 0.09333403980256323], g = [0.0], h = [0.0], x = [5.940e-02, 7.688e-01, …, 4.927e-01])
 (f = [1.0012768415285382, 0.06341488869494405, 0.0028311904366864252], g = [0.0], h = [0.0], x = [1.796e-03, 4.027e-02, …, 5.375e-01])
 (f = [5.844138530457289e-17, 0.9544202515412942, 0.3021136895136588], g = [0.0], h = [0.0], x = [1.952e-01, 1.000e+00, …, 4.944e-01])
 (f = [3.656008541215839e-17, 0.5970715056392251, 0.8065540684662647], g = [0.0], h = [0.0], x = [5.943e-01, 1.000e+00, …, 4.955e-01])
 (f = [0.7816133002621791, 0.004158493586445737, 0.6285397101440477], g = [0.0], h = [0.0], x = [4.312e-01, 3.387e-03, …, 4.845e-01])
 (f = [0.00013804384184798658, 0.9178848378551203, 0.44175878288866044], g = [0.0], h = [0.0], x = [2.856e-01, 9.999e-01, …, 5.213e-01])
 (f = [1.7391551016910975, 0.0, 0.0], g = [0.0], h = [0.0], x = [0.000e+00, 0.000e+00, …, 4.369e-02])
 ⋮
 (f = [0.9476685478616571, 0.3339328262114393, 0.08974316233559428], g = [0.0], h = [0.0], x = [5.671e-02, 2.157e-01, …, 4.803e-01])
 (f = [0.15533586754936976, 0.9313693420314089, 0.3339873286416648], g = [0.0], h = [0.0], x = [2.164e-01, 8.948e-01, …, 4.985e-01])
 (f = [0.5416089289447406, 0.6908236947763159, 0.48725055265054346], g = [0.0], h = [0.0], x = [3.226e-01, 5.767e-01, …, 4.923e-01])
 (f = [0.4679981864763092, 0.5305582750645328, 0.7214462792602528], g = [0.0], h = [0.0], x = [5.062e-01, 5.398e-01, …, 4.813e-01])
 (f = [0.20429422627398622, 0.9884556364259565, 0.008272791921642755], g = [0.0], h = [0.0], x = [5.218e-03, 8.703e-01, …, 4.890e-01])
 (f = [0.4993752594250251, 0.005988051385664177, 0.8757842282152144], g = [0.0], h = [0.0], x = [6.701e-01, 7.633e-03, …, 4.704e-01])
 (f = [0.6108433649147249, 0.7098914076836047, 0.3682171347738249], g = [0.0], h = [0.0], x = [2.385e-01, 5.477e-01, …, 4.813e-01])
 (f = [0.15492438021949514, 0.9736422510101419, 0.21880311278467507], g = [0.0], h = [0.0], x = [1.390e-01, 8.995e-01, …, 4.938e-01])
 (f = [0.4771542512137947, 0.3253039065636177, 0.8218308441853747], g = [0.0], h = [0.0], x = [6.101e-01, 3.809e-01, …, 5.153e-01])
julia> reference_point = nadir(result.population)3-element Array{Float64,1}:
 1.7391551016910975
 1.0435047298767883
 1.0021277671803095
julia> hv = hypervolume(approx_front, reference_point)1.2125592879790739

$\Delta_p$ (Delta $p$)

Metaheuristics.PerformanceIndicators.deltap — Function

deltap(front, true_pareto_front; p = 1)
Δₚ(front, true_pareto_front; p = 1)

Returns the averaged Hausdorff distance indicator aka Δₚ (Delta p).

"Δₚ" can be typed as \Delta<tab>\_p<tab>.

Parameters

front and true_pareto_front can be:

N×m matrix where N is the number of points and m is the number of objectives.
Array{xFgh_indiv} (usually State.population)

source

$\varepsilon$-Indicator

Unary and binary $\varepsilon$-indicator (epsilon-indicator). Details in E. Zitzler, L. Thiele, M. Laumanns, C.M. Fonseca, V.G. da Fonseca (2003)

Metaheuristics.PerformanceIndicators.epsilon_indicator — Function

epsilon_indicator(A, B)

Computes the ε-indicator for non-dominated sets A and B. It is assumed that all values in A and B are positive. If negative, the sets are translated to positive values.

Interpretation

epsilon_indicator(A, PF) is unary if PF is the Pareto-optimal front.
epsilon_indicator(A, B) == 1 none is better than the other.
epsilon_indicator(A, B) < 1 means that A is better than B.
epsilon_indicator(A, B) > 1 means that B is better than A.
Values closer to 1 are preferable.

Examples

julia> A1 = [4 7;5 6;7 5; 8 4.0; 9 2];

julia> A2 = [4 7;5 6;7 5; 8 4.0];

julia> A3 = [6 8; 7 7;8 6; 9 5;10 4.0 ];

julia> PerformanceIndicators.epsilon_indicator(A1, A2)
1.0

julia> PerformanceIndicators.epsilon_indicator(A1, A3)
0.9

julia> f, bounds, pf = Metaheuristics.TestProblems.ZDT3();

julia> res = optimize(f, bounds, NSGA2());

julia> PerformanceIndicators.epsilon_indicator(res, pf)
1.00497701620997

source