% Please rename the file to the function name before use.
% Tested on MATLAB R2013a
% -------------------------------------------------------------------------
% Function:  [TrueLrr,MeanLrrSims,StdLrrSims, MeanLrrApprox,StdLrrApprox] =
% meanAndStdLForSubSampling( points, area, samplingRatio ,r)
% -------------------------------------------------------------------------
% Aim: Estimate mean L(r)-r function and the standard deviation, if a set of
% (true) points are subsampled by a given sampling ratio, based on the
% analytical method presented in the paper
% -------------------------------------------------------------------------
% Input: points - X and Y coordinates as column vectors (size(points,2)=2)
% area - area of analysis
% samplingRatio - What fraction of the points should be sampled
% between 0 and 1; 
% r - radius values at which the estimates are to be made(array).
% -------------------------------------------------------------------------
% Output: 
% TrueLrr - true L(r)-r function  (all points)
% MeanLrrSims - mean L(r)-r function for the subsampled points (simulations) 
% StdLrrSims - std in L(r)-r function for the subsampled points (simulations) 
% MeanLrrApprox - mean L(r)-r function for the subsampled points (analytical) 
% StdLrrApprox - std in L(r)-r function for the subsampled points (analytical) 
% -------------------------------------------------------------------------
% Comments: The code uses a fast integral approximation of the method
% presented in the paper, using binomial theorem, to obtain MeanLrrApprox
% and StdLrrApprox. For the method presented in the paper itself(estimateWithBruteMethod), please
% uncomment the relevant lines. However, nchoosek runs into computational
% problems when number of points become large.
% -------------------------------------------------------------------------
% Written by Arun Shivanandan and Jayakrishnan Unnikrishnan EPF Lausanne,
% Last updated 9 Dec 2014 by Arun Shivanandan.


function [TrueLrr,MeanLrrSims,StdLrrSims, MeanLrrApprox,StdLrrApprox]=meanAndStdLForSubSampling( points, area, samplingRatio ,r)
numSims=100;
DIST = calcDist(points);
L=calcLfunction(DIST,r,area);
TrueLrr=L-r';
figure; hold on;
plot(r, TrueLrr,'magenta');
[MeanLrrSims, StdLrrSims] = estimateWithSimulations( points, area, samplingRatio ,r,numSims);
errorbar(r,MeanLrrSims,StdLrrSims,'green');
[MeanLrrApprox, StdLrrApprox] = estimateWithApproxMethod( points, area, samplingRatio ,r);
errorbar(r,MeanLrrApprox,StdLrrApprox,'red');
legend('True points','Simulated subsampling','Approx. analytical');

%[MeanLrrBrute, StdLrrBrute] = estimateWithBruteMethod( points, area,
%samplingRatio ,r);
%errorbar(r,MeanLrrBrute,StdLrrBrute,'blue');
%legend('True points','Simulated subsampling','Approx. analytical','Exact K. analytical');

end

function [MeanLrrSims, StdLrrSims] = estimateWithSimulations( points, area, samplingRatio ,r,numSims)

LObserved = zeros(numSims,length(r));
for k=1:numSims
    subsampledPoints = subsamplePointPattern(points, samplingRatio );
    DIST = calcDist(subsampledPoints);
    LObserved(k,:)=calcLfunction(DIST,r,area);
end

MeanLrrSims=mean(LObserved)-r;
StdLrrSims=std(LObserved);

end

function [ MeanLrrApprox, StdLrrApprox] = estimateWithApproxMethod( points, area, samplingRatio ,r)

DIST = calcDist(points);
N=size(DIST,1);
beta=samplingRatio/(1-samplingRatio);
alphaN=(1-samplingRatio)^N;
alpha=1-samplingRatio;


funM1 = @(x,N) (((1+x)*alpha).^N );
fun = @(x,N) ((((1+x)*alpha).^N - alphaN)./x);
fun1 = @(x,a, N) (((((1+x)*alpha).^N - alphaN)./(x)) .* (x <= a) .* (1./a));
fun2= @(x,a,b, N) (((((1+x)*alpha).^N - alphaN)./(x.*a.*b)) .* (x <= a) .* (a <= b));


qm1=funM1(beta,N)-alphaN; %computes S0
q0 = integral(@(x)fun(x,N),eps,beta); %computes S1
q1= integral2(@(x,a)fun1(x,a,N), eps, beta, eps, beta,'Method','iterated',...
    'AbsTol',.0001,'RelTol',.0001); %computes S2
q2= integral3(@(x,a,b)fun2(x,a,b,N), eps, beta, eps, beta,eps, beta,'Method','iterated',...
    'AbsTol',.0001,'RelTol',.001); %computes S3

Once=1 * q2 /N;
TwiceForMean=1/(N*(N-1)) * (qm1-q0);
Twice= 1/(N*(N-1)) * (q1-q2);
Thrice= 1/(N*(N-1)*(N-2)) * (q0-3*q1 + 2*q2);
Fourth = 1 / (N*(N-1)*(N-2)*(N-3)) * (qm1-6*q0 +11*q1 -6*q2);

tt=zeros(1,length(r));
ss=zeros(1,length(r));

for ijk = 1:length(r)
    B = DIST < r(ijk);
    B = B.*(1-eye(size(B)));
    tt(ijk)= sum(sum(B));
    a3= sum(sum(B).^2);
    D =triu(DIST < r(ijk));
    D = D.*(1-eye(size(D)));
    a1 = sum(D(:))^2;
    a2 = sum(D(:));
    ss1 = (a3 )*(Thrice-Fourth) + a1*(Fourth ) + a2*(Twice - 2*Thrice + Fourth);
    ss(ijk)=4*ss1;
end

ExpK = tt.'*TwiceForMean*area;
ExpKsq =ss.';
var1 =ExpKsq;
var=(var1(:)/((1*1/(area*1.0))*(1*1/(area*1.0))))-ExpK.*ExpK;
MeanK=ExpK;
StdK=sqrt(var);
MeanLrrApprox=sqrt((MeanK)/pi)-r';
StdLrrApprox=sqrt((MeanK+StdK)/pi)-sqrt((MeanK)/pi);

end

function [ DIST ] = calcDist( Points )
A = bsxfun(@minus,transpose(Points),mean(transpose(Points),2));
S = full(dot(A,A,1));
DIST = sqrt(bsxfun(@plus,S,S')-full(2*(A'*A)));
end

function [MeanLrrBrute, StdLrrBrute] = estimateWithBruteMethod( points, area, samplingRatio ,r)
NoOfPts=size(points,1);
ExpKsq=zeros(length(r),1);
DIST = calcDist(points);
%===for variance===
fourtimes=0;
thrice=0;
twice=0;
once = 0;
%==============

twiceformean = 0;%===for mean===
for kk = 1:NoOfPts
    oncenow = nchoosek(NoOfPts,kk)*samplingRatio^kk*(1-samplingRatio)^(NoOfPts - kk)/kk^4*kk/(NoOfPts);
    twiceformean = twiceformean  + nchoosek(NoOfPts,kk)*samplingRatio^kk*(1-samplingRatio)^(NoOfPts - kk)/kk^2*kk/(NoOfPts)*(kk-1)/(NoOfPts-1);
    twicenow = oncenow*(kk-1)/(NoOfPts-1);
    thricenow = twicenow*(kk-2)/(NoOfPts-2);
    fourtimesnow = thricenow*(kk-3)/(NoOfPts-3);
    once = once + oncenow;
    twice= twice + twicenow;
    thrice = thrice+thricenow;
    fourtimes = fourtimes + fourtimesnow;
end

two =0;
three = 0;
four = 0;
one = 0;
count=0;
ExpK = zeros(length(r),1);

for i=1:NoOfPts
    for j=1:NoOfPts
        if i == j
            continue
        end
        t=(DIST(i,j)<r(:));
        ExpK = ExpK + twiceformean*t;
        for k=1:NoOfPts
            for l=1:NoOfPts
                if k == l
                    continue
                end
                count=count+1;
                s=length(unique([i j k l]));  % how many are not equal
                
                
                
                t=(DIST(i,j)<r(:)).*(DIST(k,l)<r(:));
                
                if(s==4)
                    ExpKsq=ExpKsq+fourtimes*t;
                    four=four+1;
                else
                    if(s==3)
                        ExpKsq=ExpKsq+thrice*t;
                        three=three+1;
                    else
                        if(s==2)
                            ExpKsq=ExpKsq+twice*t;
                            two=two+1;
                        else
                            ExpKsq=ExpKsq+once*t;
                            one=one+1;
                        end
                    end
                end
                
                
            end
        end
    end
    
end

ExpK = ExpK*area;


var1 =ExpKsq;
var=var1(:)/((1*1/(area*1.0))*(1*1/(area*1.0)))-ExpK.*ExpK; %Var=Exp(K^2)-Exp(K)^2
stdk=sqrt(var);
MeanLrrBrute=sqrt((ExpK)/pi)-r';
StdLrrBrute=sqrt((ExpK+stdk)/pi)-sqrt((ExpK)/pi);

end




function L=calcLfunction(DIST,r,area)

L=sqrt(calcKfunction(DIST,r,area)/pi);

end

function [K]=calcKfunction(DIST, r, area)

K = zeros(length(r),1);
for k=1:length(K)
    K(k) = sum(sum(DIST<r(k)));
end
K=K-size(DIST,1);
K=K/(size(DIST,1)*size(DIST,1)/(area*1.0));

end

function [ subsampledPoints ] = subsamplePointPattern( points, samplingRatio )

indices=rand(size(points,1),1)<samplingRatio;
subsampledPoints=points(indices,:);

end