Smoothing measured data in MATLAB? - algorithm

Question

I have measured data from MATLAB and I'm wondering how to best smooth the data?
Example data (1st colum=x-data / second-colum=y-data):
33400 209.11
34066 210.07
34732 212.3
35398 214.07
36064 215.61
36730 216.95
37396 218.27
38062 219.52
38728 220.11
39394 221.13
40060 221.4
40726 222.5
41392 222.16
42058 223.29
42724 222.77
43390 223.97
44056 224.42
44722 225.4
45388 225.32
46054 225.98
46720 226.7
47386 226.53
48052 226.61
48718 227.43
49384 227.84
50050 228.41
50716 228.57
51382 228.92
52048 229.67
52714 230.02
53380 229.54
54046 231.19
54712 231.00
55378 231.5
56044 231.5
56710 231.79
57376 232.26
58042 233.12
58708 232.65
59374 233.51
60040 234.16
60706 234.21
The data in the second column should be monoton but it isn't. How to make it smooth?
I could probably invent a short algorithm myself but I think it's a better way to use an established and proven one... do you know a good way to somehow integrate the outliners to make it a monoton curve!?
Thanks in advance

Answer 1

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.