UN4 - 1 year ago 233

Python Question

I would like to fit my surface equation to some data. I already tried scipy.optimize.leastsq but as I cannot specify the bounds it gives me an unusable results. I also tried scipy.optimize.least_squares but it gives me an error:

`ValueError: too many values to unpack`

My equation is:

`f(x,y,z)=(x-A+y-B)/2+sqrt(((x-A-y+B)/2)^2+C*z^2)`

parameters A, B, C should be found so that the equation above would be as close as possible to zero when the following points are used for x,y,z:

`[`

[-0.071, -0.85, 0.401],

[-0.138, -1.111, 0.494],

[-0.317, -0.317, -0.317],

[-0.351, -2.048, 0.848]

]

The bounds would be A > 0, B > 0, C > 1

How I should obtain such a fit? What is the best tool in python to do that. I searched for examples on how to fit 3d surfaces but most of examples involving function fitting is about line or flat surface fits.

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Answer Source

This isn't strictly least squares, but how about something like this?

```
import numpy as np
import scipy.optimize
coeff_0 = np.array([1, 1, 1])
coeff = coeff_0
data = np.array([
[-0.071, -0.85, 0.401],
[-0.138, -1.111, 0.494],
[-0.317, -0.317, -0.317],
[-0.351, -2.048, 0.848]
])
def objective(coeff, data):
# Function that returns the squared loss.
# We want the function to choose A, B, C such that all values are close to zero
A, B, C = coeff
x, y, z = data.T
objective = (x - A + y - B) / 2 + np.sqrt(((x - A - y + B) / 2) ** 2 + C * z**2)
# L2 regularization
reg = .01 * np.sqrt((coeff ** 2).sum())
value = objective + reg
losses = (value - 0) ** 2
loss = losses.mean()
return loss
result = scipy.optimize.minimize(objective, coeff_0, args=(data,))
coeff = result.x
A, B, C = result.x
loss = objective(result.x, data)
print('final loss = %r' % (loss,))
```

This solution is like throwing a sledge hammer at the problem. There probably is a way to use least squares to get a solution more efficiently using an SVD solver, but if you're just looking for an answer BFGS optimization will find you one.

I found in the documentation that all you need to do to adapt this to actual least squares is to specify the function that computes the residuals.

```
def target_func(A, B, C, x, y, z):
return (x - A + y - B) / 2 + np.sqrt(((x - A - y + B) / 2) ** 2 + C * z ** 2)
def residuals(coeff, data):
# Function that returns the squared loss.
# We want the function to choose A, B, C such that all values are close to zero
A, B, C = coeff
x, y, z = data.T
# The function we care about
objective = target_func(A, B, C, x, y, z)
losses = (objective - 0)
return losses
lst_sqrs_result = scipy.optimize.least_squares(residuals, coeff_0, args=(data,),
bounds=([0, 0, 0],
[np.inf, np.inf, np.inf]))
print('lst_sqrs_result = %r' % (result,))
```

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