How to make 4D scatter bubble plot marker size legend?

I am looking to display results from a multi-objective optimization with four objective functions. The 'scatter' function works nicely to display the data, but I cannot figure out how to make a proper legend for the circle sizes. Here's an example script for what I am trying to do:

%4D Bubble Plot Script
%Takes 4 Objective function values and makes a bubble plot to visualize the
%data

clear
clc
close all

%Pareto Values
x1=10*rand(100,1);
x2=10*rand(100,1);
x3=10*rand(100,1);
x4=10*rand(100,1);

%Sort According to decreasing diameter
[x4,I]=sort(x4);
x4=wrev(x4);
x1=wrev(x1(I));
x2=wrev(x2(I));
x3=wrev(x3(I));

%Generate Figure
figure()
scatter (x1,x2,100*x4,x3,'filled','MarkerEdgeColor',[0 0 0],'LineWidth',0.5)
colormap jet
c=colorbar('Limits',[0 10],'Ticks',0:10,'FontName','Times New Roman','FontSize',14);
c.Label.String = 'x_{3}   [units]';
c.Label.FontSize=14;
axis([0 10 0 10])
xlabel('x_{1}   [units]')
ylabel('x_{2}   [units]')
xticks(0:10)
yticks(0:10)
set(gca,'FontName','Times New Roman','FontSize',14,'LineWidth',1.5);
set(gcf,'color','w');

I would like to make a legend like the one I made in inkscape as an example in this figure. For the fourth variable x_4, I added a scaling factor in the 'scatter' function to make the circles appear slightly larger, so I would also like to be able to account for that in the legend if possible.

Also, when the figure is exported as an emf the '[units]' gets put into the subscript for both vertical axes, but the legend labels appear correctly in the MATLAB figure though. Any ideas how to correct that?

Answer 1

[Walter Roberson]

legend() will only create one entry for each plot handle, but scatter() only creates one plot handle per call.

Work around:

Lh(1) = scatter(nan, nan, min(x4)*100, 'k'); 
Lh(2) = scatter(nan, nan, max(x4)*100, 'k');
legend(Lh, {'min {x4}', 'max {x4}'} )