## How to find the best fit sine wave to some data

clear all
tic;
global x1;
global y;

## First lets make some data.
## One example of where you might get data like this: 
## If you look at Jupiter's moon's and plot their position over
## several days you can get data that is part of one sine wave.

x1=0:7;
f=.05
y=1.7*sin(2*pi*f*x1+.3);
plot(x1,y)

## now define a function that has one output
## that should be minimized.


function fs=frs(x3)
 global y;
 global x1;
  a=x3(1); # magnitude of the sine wave
  f=x3(2); # frequency
  th=x3(3); # phase shift 
## Now calculate a guess wave
 yy=a.*sin(2*pi*f*x1+th);
## find the sum of the magnitude of the errors 
 fs=sum(sum(abs(yy-y)));
endfunction

## q is the starting guess 
## [ magnitude, frequency, phase shift ]
 q=[1,.0001,.006]
## next there are 2 solvers shown -- just try either one of them
#p=fminunc(@frs,q ) 
p=sqp(q,@frs)
# The answer come back in p

## now plot the results against the original.
t=0:.01:7;
yt=p(1).*sin(2*pi*p(2).*t+p(3));
hold on
plot(t,yt,'r') # the result of the curve fitting  is the red line
hold off
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