# Trouble with Envelope Functions

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Si on 21 Apr 2015
Commented: Hany Ferdinando on 29 Jan 2019
Hi,
I have been trying to obtain a smooth envelope of my data. I have tried using other peoples solutions from the file exchange but unable to get a smooth envelope. See attached images.
Attached is the relevant data.
If anyone can recommend a file exchange or other solution would be much appreciated!
Thanks!

#### 1 Comment

Glo on 21 Apr 2015
Can you be more specific about your question? What is this data? What do you mean by "smooth envelope"? What is the specific goal?

John D'Errico on 21 Apr 2015
Edited: John D'Errico on 21 Apr 2015
The problem is, you don't really want an envelope.
For example, here are a couple of fits that will produce an envelope.
slm_upper = slmengine(x,y,'env','sup','plot','on','knots',30);
slm_lower = slmengine(x,y,'env','inf','plot','on','knots',30);
Those are envelope fits, i.e., least upper bound and greatest lower bound functions.
But what you have drawn are functions that sort of look like that, but go where you want them to go, ignoring some of your data. So they skip some points that you consider outliers. You want maybe a function that can intelligently (defined by what you consider an outlier) exclude perhaps some 1 to 5% of the points as outliers.
The problem is, the eye is good at seeing a pattern that it likes. The computer, not so good. Computers do what they are programmed to do.
Perhaps the best suggestion I can offer is to use a tool like my SLM as below in a multiple step process:
slm_upper0 = slmengine(x,y,'env','sup','plot','on','knots',30);
tol= max(y)*1.e-14;
res_upper = slmeval(x,slm_upper,0) - y;
keep_upper = find(res_upper > tol);
drop_upper = find(res_upper <= tol);
slm_upper = slmengine(x(keep_upper),y(keep_upper),'env','sup','plot','on','knots',30);
hold on
plot(x(drop_upper),y(drop_upper),'ro')
Thus, find the points that were on the boundary in one envelope, then exclude them form the fit, and repeat the fit. Do a similar procedure for a lower semi-envelope. I'm not sure I see a better way.
The SLM toolbox is on the file exchange.

#### 1 Comment

Si on 21 Apr 2015

Youssef Khmou on 21 Apr 2015
Edited: Youssef Khmou on 21 Apr 2015
try the following basic solution using Hilbert transform :
fs=40;
t=0:1/fs:4-1/fs;
f=15;
y=1.5*sin(2*pi*f*t).*exp(-1.1*t);
y=y+0.1*randn(size(t));
plot(t,y)
hold on;
z=abs(hilbert(y));
plot(t,smooth(z,0.25),'r');