Assume one has data, which for sake of convenience we will generate in Matlab:
x=[0:10]
y=exp(-.7*x)

xx=x'
yy=y'

plot(x,y)

But we did not really know where it came from

Plotting it suggests an exponential decay, but we do not know the rate- i.e. the .7.


we would like to find u such that

z=|| exp(-u*xx)-yy|| is minimized so we will minimize the error.

u occurs nonlinearly!

There is a 1 dimensional search engine fminbnd that takes a function of 1 variable and bounds and produces a minimum


so we write a function which given u, xx and yy will return z.

Open the editor and type

function z=exp3(u,xx,yy)
z=norm(exp(-u*xx)-yy);
disp([u z])


save this as exp3.

Note that exp3 has 3 variables.

we can get around this by the following trick

[zz,FVAL,EXITFLAG,OUTPUT] =fminbnd(@(u)exp3(u,xx,yy),.1,2)

which says find me u between .1 and 2

try this.

now look at data which might have more than 1 variable

what if we had generated data as
y = 2*exp(-.1*x)+ 3*exp(-1.5*x)
and set yy=y'

assume we thought perhaps there were 4 parameters- weighted sum of 2 exponentials but we did not really know  the 2 , -1,3,1.5

Create the function using the editor
function z=exp4(u,xx,yy)
z=norm(u(1)*exp(-u(2)*xx)+ u(3)*exp(-u(4)*xx)-yy);
disp([u(1) u(2) u(3) u(4) z])

and save as exp4

now use fminsearch
[zz,FVAL,EXITFLAG,OUTPUT]=fminsearch(@(v)exp4(v,xx,yy),[1;2;3;4])
and stand back

there is also one especially for nonlinear functions as in:
function zz=exp5(u,xx,yy)
zz=u(1)*exp(-u(2)*xx)+ u(3)*exp(-u(4)*xx)-yy;
disp([u(1) u(2) u(3) u(4) ])

calling
[zz,FVAL,EXITFLAG,OUTPUT]=lsqnonlin(@(v)exp5(v,xx,yy),[1;2;3;4])