布谷鸟哈希函数的参数_布谷鸟算法详细讲解
今天我要講的內(nèi)容是布谷鳥算法，英文叫做Cuckoo search (CS algorithm)。首先還是同樣，介紹一下這個算法的英文含義， Cuckoo是布谷鳥的意思，啥是布谷鳥呢，是一種叫做布谷的鳥，o(∩_∩)o ，這種鳥她媽很懶，自己生蛋自己不養(yǎng)，一般把它的寶寶扔到別的種類鳥的鳥巢去。但是呢，當孵化后，遇到聰明的鳥媽媽，一看就知道不是親生的，直接就被鳥媽媽給殺了。于是這群布谷鳥寶寶為了保命，它們就模仿別的種類的鳥叫，讓智商或者情商極低的鳥媽媽誤認為是自己的親寶寶，這樣它就活下來了。 Search指的是搜索，這搜索可不是谷歌一下，你就知道。而是搜索最優(yōu)值，舉個簡單的例子，y=(x-0.5)^2+1，它的最小值是1，位置是(0.5,1)，我們要搜索的就是這個位置。
現(xiàn)在我們應該清楚它是干嘛的了吧，它就是為了尋找最小值而產(chǎn)生的一種算法，有些好裝X的人會說，你傻X啊，最小值不是-2a/b嗎，用你找啊? 說的不錯，確實是，但是要是我們的函數(shù)變成 y=sin(x^3+x^2)+e^cos(x^3)+log(tan(x)+10，你怎么辦吶？你解不了，就算你求導數(shù)，但是你知道怎么解導數(shù)等于0嗎？所以我們就得引入先進的東西來求最小值。
為了使大家容易理解，我還是用y=(x-0.5)^2+1來舉例子，例如我們有4個布谷鳥蛋(也就是4個x坐標)，鳥媽媽發(fā)現(xiàn)不是自己的寶寶的概率是0.25，我們x的取值范圍是[0,1]之間，于是我們就可以開始計算了。
目標：求x在[0,1]之內(nèi)的函數(shù)y=(x-0.5)^2+1最小值
(1)初始化x的位置，隨機生成4個x坐標，x1=0.4，x2=0.6，x3=0.8，x4=0.3 ——> X=[0.4, 0.6 ,0.8, 0.3]
(2)求出y1~y4，把x1~x4帶入函數(shù)，求得Y=[1,31, 1.46, 1.69, 1.265]，并選取當前最小值ymin= y4=1.265
(3)開始定出一個y的最大值為Y_global=INF(無窮大)，然后與ymin比較，把Y中最小的位置和值保留，例如Y_global=INF>ymin=1.265，所以令Y_global=1.265
(4)記錄Y_global的位置，(0.3，1.265)。
(5)按概率0.25，隨機地把X中的值過塞子，選出被發(fā)現(xiàn)的蛋。例如第二個蛋被發(fā)現(xiàn)x2=0.6，那么他就要隨機地變換位子，生成一個隨機數(shù)，例如0.02，然后把x2=x2+0.02=0.62，之后求出y2=1.4794。那么X就變?yōu)榱薠=[0.4, 0.62 ,0.8, 0.3]，Y=[1,31, 1.4794, 1.69, 1.265]。
(6)進行萊維飛行，這名字聽起來挺高大上，說白了，就是把X的位置給隨機地改變了。怎么變？有一個公式x=x+alpha*L。
L=S*(X-Y_global)*rand3
S=[rand1*sigma/|rand2|]^(1/beta)
sigma=0.6966
beta=1.5
alpha=0.01
rand1~rand3為正態(tài)分布的隨機數(shù)
然后我們把X=[0.4, 0.6 ,0.8, 0.3]中的x1帶入公式，首先隨機生成rand1=-1.2371，rand2=-2.1935，rand3=-0.3209，接下來帶入公式中，獲得x1=0.3985
之后同理計算:
x2=0.6172
x3=0.7889
x4=0.3030
(7)更新矩陣X，X=[0.3985, 0.6172, 0.7889, 0.3030]
(8)計算Y=[1.3092, 1.4766, 1.6751, 1.2661]，并選取當前最小值ymin= y4=1.2661，然后與ymin比較，把Y中最小的位置和值保留，例如Y_global=1.265
(9)返回步驟(5)用更新的X去循環(huán)執(zhí)行，經(jīng)過多次計算即可獲得y的最優(yōu)值和的最值位置(x,y)
代碼：
% -----------------------------------------------------------------
% Cuckoo Search (CS) algorithm by Xin-She Yang and Suash Deb %
% Programmed by Xin-She Yang at Cambridge University %
% Programming dates: Nov 2008 to June 2009 %
% Last revised: Dec 2009 (simplified version for demo only) %
% -----------------------------------------------------------------
% Papers --Citation Details:% 1) X.-S. Yang, S. Deb, Cuckoo search via Levy flights,% in: Proc. of World Congress on Nature &Biologically Inspired% Computing (NaBIC 2009), December 2009, India,% IEEE Publications, USA, pp. 210-214 (2009).% http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.1594v1.pdf
% 2) X.-S. Yang, S. Deb, Engineering optimization by cuckoo search,%Int. J. Mathematical Modelling and Numerical Optimisation,% Vol. 1, No. 4, 330-343 (2010).% http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2908v2.pdf
% ----------------------------------------------------------------%
% This demo program only implements a standard version of %
% Cuckoo Search (CS), as the Levy flights and generation of %
% new solutions may use slightly different methods. %
% The pseudo code was given sequentially (select a cuckoo etc), %
% but the implementation here uses Matlab's vector capability, %
% which results in neater/better codes and shorter running time. %
% This implementation is different and more efficient than the %
% the demo code provided inthe book by% "Yang X. S., Nature-Inspired Metaheuristic Algoirthms, %
% 2nd Edition, Luniver Press, (2010). "%
% --------------------------------------------------------------- %
% =============================================================== %
% Notes: %
% Different implementations may lead to slightly different %
% behavour and/or results, but there is nothing wrong with it, %
% as this is the nature of random walks and all metaheuristics. %
% -----------------------------------------------------------------
% Additional Note: This version uses a fixed number of generation %
% (not a given tolerance) because many readers asked me to add %
% or implement this option. Thanks.%function [bestnest,fmin]=cuckoo_search_new(n)if nargin<1,%Number of nests (or different solutions)
n=25;
end% Discovery rate of alien eggs/solutions
pa=0.25;%% Change this if you want to getbetter results
N_IterTotal=1000;%%Simple bounds of the search domain%Lower bounds
nd=15;
Lb=-5*ones(1,nd);%Upper bounds
Ub=5*ones(1,nd);%Random initial solutionsfor i=1:n,
nest(i,:)=Lb+(Ub-Lb).*rand(size(Lb));
end%Get the current best
fitness=10^10*ones(n,1);
[fmin,bestnest,nest,fitness]=get_best_nest(nest,nest,fitness);
N_iter=0;%%Starting iterationsfor iter=1:N_IterTotal,% Generate newsolutions (but keep the current best)
new_nest=get_cuckoos(nest,bestnest,Lb,Ub);
[fnew,best,nest,fitness]=get_best_nest(nest,new_nest,fitness);%Update the counter
N_iter=N_iter+n;%Discovery and randomization
new_nest=empty_nests(nest,Lb,Ub,pa) ;% Evaluate this setof solutions
[fnew,best,nest,fitness]=get_best_nest(nest,new_nest,fitness);%Update the counter again
N_iter=N_iter+n;%Find the best objective so farif fnew
fmin=fnew;
bestnest=best;
end
end%%End of iterations%% Post-optimization processing%%Display all the nests
disp(strcat('Total number of iterations=',num2str(N_iter)));
fmin
bestnest%% --------------- All subfunctions are list below ------------------
%%Get cuckoos by ramdom walk
function nest=get_cuckoos(nest,best,Lb,Ub)%Levy flights
n=size(nest,1);%Levy exponent and coefficient% For details, see equation (2.21), Page 16 (chapter 2) of the book% X. S. Yang, Nature-Inspired Metaheuristic Algorithms, 2nd Edition, Luniver Press, (2010).
beta=3/2;
sigma=(gamma(1+beta)*sin(pi*beta/2)/(gamma((1+beta)/2)*beta*2^((beta-1)/2)))^(1/beta);for j=1:n,
s=nest(j,:);% This isa simple way of implementing Levy flights% For standard random walks, use step=1;%% Levy flights by Mantegna's algorithm
u=randn(size(s))*sigma;
v=randn(size(s));
step=u./abs(v).^(1/beta);% In the next equation, the difference factor (s-best) means that% when the solution isthe best solution, it remains unchanged.
stepsize=0.01*step.*(s-best);% Here the factor 0.01 comes from the fact that L/100should the typical% step size of walks/flights where L isthe typical lenghtscale;% otherwise, Levy flights may become too aggresive/efficient,% which makes new solutions (even) jump outside of the design domain%(and thus wasting evaluations).%Now the actual random walks or flights
s=s+stepsize.*randn(size(s));% Apply simple bounds/limits
nest(j,:)=simplebounds(s,Lb,Ub);
end%%Find the current best nest
function [fmin,best,nest,fitness]=get_best_nest(nest,newnest,fitness)% Evaluating all newsolutionsfor j=1:size(nest,1),
fnew=fobj(newnest(j,:));if fnew<=fitness(j),
fitness(j)=fnew;
nest(j,:)=newnest(j,:);
end
end%Find the current best
[fmin,K]=min(fitness) ;
best=nest(K,:);%% Replace some nests by constructing new solutions/nests
function new_nest=empty_nests(nest,Lb,Ub,pa)%A fraction of worse nests are discovered with a probability pa
n=size(nest,1);% Discovered or not --a status vector
K=rand(size(nest))>pa;% In the real world, if a cuckoo's egg is very similar to a host's eggs, then% this cuckoo's egg is less likely to be discovered, thus the fitness should
% be related to the difference in solutions. Therefore, it isa good idea% to do a random walk ina biased way with some random step sizes.%% New solution by biased/selective random walks
stepsize=rand*(nest(randperm(n),:)-nest(randperm(n),:));
new_nest=nest+stepsize.*K;for j=1:size(new_nest,1)
s=new_nest(j,:);
new_nest(j,:)=simplebounds(s,Lb,Ub);
end%Application of simple constraints
function s=simplebounds(s,Lb,Ub)%Apply the lower bound
ns_tmp=s;
I=ns_tmp
ns_tmp(I)=Lb(I);%Apply the upper bounds
J=ns_tmp>Ub;
ns_tmp(J)=Ub(J);% Update this newmove
s=ns_tmp;%%You can replace the following by your own functions% A d-dimensional objective function
function z=fobj(u)%% d-dimensional sphere function sum_j=1^d (u_j-1)^2.% with a minimum at (1,1, ...., 1);
z=sum((u-1).^2);
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