MATLAB算法应用案例精讲-智能优化算法多目标算法性能评价指标

前言

在评估多目标优化算法的效果时，我们通常使用五个主要指标：GD（Generational Distance）、IGD（Inverted Generational Distance）、Hypervolume、Spacing和Spread。

GD和IGD是用于测量算法生成的解集合与真实前沿解之间的距离的指标。具体而言，GD测量了所有生成解与真实前沿解之间的平均欧几里得距离，而IGD测量了所有真实前沿解与生成解之间的平均欧几里得距离。这两个指标的目标是越小越好，即算法的生成解集合应该尽可能接近真实前沿解。

Hypervolume则是用于测量算法生成的解集合能够覆盖的真实前沿解的体积大小的指标。目标是越大越好，即算法生成的解集合应该尽可能覆盖更多的真实前沿解。

Spacing和Spread是用于测量算法生成的解集合的分布情况的指标。Spacing测量了所有解之间的最小欧几里得距离，而Spread测量了所有解之间的最大欧几里得距离。

通过对这些指标的全面评估，我们可以更好地了解算法的表现并进行优化。

代码实现

MATLAB

Spread.m

1.  
   %% PopObj：算法求得的pareto解集
2.  
   %% PF：真实的解集
3.  
   function Score = Spread(PopObj,PF)
4.  
   Dis1 = pdist2(PopObj,PopObj);
5.  
   Dis1(logical(eye(size(Dis1,1)))) = inf;
6.  
   [~,E] = max(PF,[],1);PF(E,:)
7.  
   Dis2 = pdist2(PF(E,:),PopObj);
8.  
   d1 = sum(min(Dis2,[],2));
9.  
   d2 = mean(min(Dis1,[],2));
10.  
    Score = (d1 sum(abs(min(Dis1,[],2)-d2))) / (d1 (size(PopObj,1)-size(PopObj,2))*d2);
11.  
    end

Spacing.m

1.  
   %% PopObj：算法求得的pareto解集
2.  
   %% PF：真实的解集
3.  
   function Score = Spacing(PopObj,PF)
4.  
   Distance = pdist2(PopObj,PopObj,'cityblock');
5.  
   Distance(logical(eye(size(Distance,1)))) = inf;
6.  
   Score = std(min(Distance,[],2));
7.  
   end

IGD.m

1.  
   %% PopObj：算法求得的pareto解集
2.  
   %% PF：真实的解集
3.  
   function Score = IGD(PopObj,PF)
4.  
   Distance = min(pdist2(PF,PopObj),[],2);
5.  
   Score = mean(Distance);
6.  
   end

HV.m

1.  
   %% PopObj：算法求得的pareto解集
2.  
   %% PF：真实的解集
3.  
   function [Score,PopObj] = HV(PopObj,PF)
4.  
   % Hypervolume
5.  
   [N,M] = size(PopObj);
6.  
   fmin = min(min(PopObj,[],1),zeros(1,M));
7.  
   fmax = max(PF,[],1);
8.  
   PopObj = (PopObj-repmat(fmin,N,1))./repmat((fmax-fmin)*1.1,N,1);
9.  
   PopObj(any(PopObj>1,2),:) = [];
10.  
    % The reference point is set to (1,1,...)
11.  
    RefPoint = ones(1,M);
12.  
    if isempty(PopObj)
13.  
    Score = 0;
14.  
    elseif M < 4
15.  
    % Calculate the exact HV value
16.  
    pl = sortrows(PopObj);
17.  
    S = {1,pl};
18.  
    for k = 1 : M-1
19.  
    S_ = {};
20.  
    for i = 1 : size(S,1)
21.  
    Stemp = Slice(cell2mat(S(i,2)),k,RefPoint);
22.  
    for j = 1 : size(Stemp,1)
23.  
    temp(1) = {cell2mat(Stemp(j,1))*cell2mat(S(i,1))};
24.  
    temp(2) = Stemp(j,2);
25.  
    S_ = Add(temp,S_);
26.  
    end
27.  
    end
28.  
    S = S_;
29.  
    end
30.  
    Score = 0;
31.  
    for i = 1 : size(S,1)
32.  
    p = Head(cell2mat(S(i,2)));
33.  
    Score = Score cell2mat(S(i,1))*abs(p(M)-RefPoint(M));
34.  
    end
35.  
    else
36.  
    % Estimate the HV value by Monte Carlo estimation
37.  
    SampleNum = 1000000;
38.  
    MaxValue = RefPoint;
39.  
    MinValue = min(PopObj,[],1);
40.  
    Samples = unifrnd(repmat(MinValue,SampleNum,1),repmat(MaxValue,SampleNum,1));
41.  
    if gpuDeviceCount > 0
42.  
    % GPU acceleration
43.  
    Samples = gpuArray(single(Samples));
44.  
    PopObj = gpuArray(single(PopObj));
45.  
    end
46.  
    for i = 1 : size(PopObj,1)
47.  
    drawnow();
48.  
    domi = true(size(Samples,1),1);
49.  
    m = 1;
50.  
    while m <= M && any(domi)
51.  
    domi = domi & PopObj(i,m) <= Samples(:,m);
52.  
    m = m 1;
53.  
    end
54.  
    Samples(domi,:) = [];
55.  
    end
56.  
    Score = prod(MaxValue-MinValue)*(1-size(Samples,1)/SampleNum);
57.  
    end
58.  
    end
59.  
     
60.  
    function S = Slice(pl,k,RefPoint)
61.  
    p = Head(pl);
62.  
    pl = Tail(pl);
63.  
    ql = [];
64.  
    S = {};
65.  
    while ~isempty(pl)
66.  
    ql = Insert(p,k 1,ql);
67.  
    p_ = Head(pl);
68.  
    cell_(1,1) = {abs(p(k)-p_(k))};
69.  
    cell_(1,2) = {ql};
70.  
    S = Add(cell_,S);
71.  
    p = p_;
72.  
    pl = Tail(pl);
73.  
    end
74.  
    ql = Insert(p,k 1,ql);
75.  
    cell_(1,1) = {abs(p(k)-RefPoint(k))};
76.  
    cell_(1,2) = {ql};
77.  
    S = Add(cell_,S);
78.  
    end
79.  
     
80.  
    function ql = Insert(p,k,pl)
81.  
    flag1 = 0;
82.  
    flag2 = 0;
83.  
    ql = [];
84.  
    hp = Head(pl);
85.  
    while ~isempty(pl) && hp(k) < p(k)
86.  
    ql = [ql;hp];
87.  
    pl = Tail(pl);
88.  
    hp = Head(pl);
89.  
    end
90.  
    ql = [ql;p];
91.  
    m = length(p);
92.  
    while ~isempty(pl)
93.  
    q = Head(pl);
94.  
    for i = k : m
95.  
    if p(i) < q(i)
96.  
    flag1 = 1;
97.  
    else
98.  
    if p(i) > q(i)
99.  
    flag2 = 1;
100.  
     end
101.  
     end
102.  
     end
103.  
     if ~(flag1 == 1 && flag2 == 0)
104.  
     ql = [ql;Head(pl)];
105.  
     end
106.  
     pl = Tail(pl);
107.  
     end
108.  
     end
109.  
      
110.  
     function p = Head(pl)
111.  
     if isempty(pl)
112.  
     p = [];
113.  
     else
114.  
     p = pl(1,:);
115.  
     end
116.  
     end
117.  
      
118.  
     function ql = Tail(pl)
119.  
     if size(pl,1) < 2
120.  
     ql = [];
121.  
     else
122.  
     ql = pl(2:end,:);
123.  
     end
124.  
     end
125.  
      
126.  
     function S_ = Add(cell_,S)
127.  
     n = size(S,1);
128.  
     m = 0;
129.  
     for k = 1 : n
130.  
     if isequal(cell_(1,2),S(k,2))
131.  
     S(k,1) = {cell2mat(S(k,1)) cell2mat(cell_(1,1))};
132.  
     m = 1;
133.  
     break;
134.  
     end
135.  
     end
136.  
     if m == 0
137.  
     S(n 1,:) = cell_(1,:);
138.  
     end
139.  
     S_ = S;
140.  
     end

GD.m

1.  
   %% PopObj：算法求得的pareto解集
2.  
   %% PF：真实的解集
3.  
   function Score = GD(PopObj,PF)
4.  
   % Generational distance
5.  
   Distance = min(pdist2(PopObj,PF),[],2);
6.  
   Score = norm(Distance) / length(Distance);
7.  
   end

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