optimisation problem

[sm2n10]

Hi, I have the following equation:


f(z)=g(z)+b*u(z)


where z=(x,y) i.e. bivariate,b is a parameter, u(z) the uniform distribution and g(z) a function that represents distance.


By considering for a momment b=0, min(f(z)) can give me the location of the minimum distance. However because I want to have locations that are not the same I add u(z). With b it's possible to change the influence of u(z). Very high values of b give very random positions, while if b is very small, only locations around the minimum are chosen.


Furthermore, I have some reference locations zr={(x1,y1),(x2,x2),...(xn,yn)}. I'm trying to figure out the best b I could have in order to produce from f(z) locations as much close as possible to zr.
Is there anything in code/method I could have in order to solve it?


Thanks

 

[Ishuda]

Hi, I have the following equation:


f(z)=g(z)+b*u(z)


where z=(x,y) i.e. bivariate,b is a parameter, u(z) the uniform distribution and g(z) a function that represents distance.


By considering for a momment b=0, min(f(z)) can give me the location of the minimum distance. However because I want to have locations that are not the same I add u(z). With b it's possible to change the influence of u(z). Very high values of b give very random positions, while if b is very small, only locations around the minimum are chosen.


Furthermore, I have some reference locations zr={(x1,y1),(x2,x2),...(xn,yn)}. I'm trying to figure out the best b I could have in order to produce from f(z) locations as much close as possible to zr.
Is there anything in code/method I could have in order to solve it?


Thanks

This is a standard model for regression fitting. That is you have a set of 'measured' data [your {(x₁,y₁), (x₂,y₂), ..., (x_n,y_n)}] which you think should follow some model [your g(z)], but the 'measurements' have some error in them [your b u(z)]. Linear regression is so called not because the model is linear in x but because the coefficients occur in a linear fashion.

As a very simple example, assume our data came from a line
x_j = x₀ + a z_j + e_xj
y_j = y₀ + b z_j + e_yj
Then the sum of squared errors E is
E = \(\displaystyle \Sigma e_{xj}^2 + e_{yj}^2\)
We want to make this as small as we can by choosing parameters x₀, y₀, a, and b. This is a simple linear regression least squares problem, see
http://mathworld.wolfram.com/LeastSquaresFitting.html
for example.

There are linear regression solvers on line, for example a selection for Google
https://www.google.com/search?q=exp...utf-8&oe=utf-8#q=linear+regression+calculator