Monotone in your case is always increasing!
See the options below (1. Cobb-Douglas; 2. Quadratic; 3. Cubic)
clear all
close all
load needSmooth.dat % Your data
x=needSmooth(:,1);
y=needSmooth(:,2);
n=length(x);
% Figure 1
logX=log(x);
logY=log(y);
Y=logY;
X=[ones(n,1),logX];
B=regress(Y,X);
a=exp(B(1,1));
b=B(2,1);
figure(1)
plot(x,y,'k*')
hold
for i=1:n-1
plot([x(i,1);x(i+1,1)],[a*x(i,1)^b;a*x(i+1,1)^b],'k-')
end
%Figure 2
X=[ones(n,1),x,x.*x];
Y=y;
B=regress(Y,X);
c=B(1,1);
b=B(2,1);
a=B(3,1);
figure(2)
plot(x,y,'k*')
hold
for i=1:n-1
plot([x(i,1);x(i+1,1)],[c+b*x(i,1)+a*x(i,1)^2; c+b*x(i+1,1)+a*x(i+1,1)^2],'k-')
end
%Figure 3
X=[ones(n,1),x,x.*x,x.*x.*x];
Y=y;
B=regress(Y,X);
d=B(1,1);
c=B(2,1);
b=B(3,1);
a=B(4,1);
figure(3)
plot(x,y,'k*')
hold
for i=1:n-1
plot([x(i,1);x(i+1,1)],[d+c*x(i,1)+b*x(i,1)^2+a*x(i,1)^3; d+c*x(i+1,1)+b*x(i+1,1)^2+a*x(i+1,1)^3],'k-')
end
There are also some cooked functions in Matlab such as "smooth" and "spline" that should also work in your case since your data is almost monotone.