Jeppe Gravgaard - 6 months ago 35

Python Question

I want to fit the function

`f(x) = b+a/x`

`leastsquares`

My code is as follows:

`x = np.asarray(range(20,401,20))`

`y = np.random.rand(20)`

`params = np.array([1,1])`

`def funcinv(x):`

return params[0]/x+params[1]

res = least_squares(funinv, params, args=(x, y))

** error given:

`return np.atleast_1d(fun(x, *args, **kwargs))`

TypeError: funinv() takes 1 positional argument but 3 were given

How can i fit my data?

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Answer Source

To make a little of clarity. There are two related problems:

- Minimizing a function
- Fitting model to data

To fit a model to observed data is to find such parameters of a model which *minimize* some sort of *error* between model data and observed data.

`least_squares`

method just minimizes a following function with respect to `x`

(`x`

can be a vector).

F(x) = 0.5 * sum(rho(f_i(x)**2), i = 0, ..., m - 1)

(`rho`

is a loss function and default is `rho(x) = x`

so don't mind it for now)

`least_squares(func, x0)`

expects that call to `func(x)`

will return a vector `[a1, a2, a3, ...]`

for which a sum of squares will be computed: `S = 0.5 * (a1^2 + a2^2 + a3^2 + ...)`

.

`least_squares`

will tweak `x0`

to minimize `S`

.

Thus, in order to use it to fit model to data, one must construct a function of error between a model and actual data - *residuals* and then minimize that *residuals* function. In your case you can write it as follows:

```
import numpy as np
from scipy.optimize import least_squares
x = np.asarray(range(20,401,20))
y = np.random.rand(20)
params = np.array([1,1])
def funcinv(x, a, b):
return b + a/x
def residuals(params, x, data):
# evaluates function given vector of params [a, b]
# and return residuals: (observed_data - model_data)
a, b = params
func_eval = funcinv(x, a, b)
return (data - func_eval)
res = least_squares(residuals, params, args=(x, y))
```

This gives a result:
`print(res)`

```
...
message: '`gtol` termination condition is satisfied.'
nfev: 4
njev: 4 optimality: 5.6774618339971994e-10
status: 1
success: True
x: array([ 6.89518618, 0.37118815])
```

However, as a residuals function pretty much the same all the time (`res = observed_data - model_data`

), there is a shortcut in `scipy.optimize`

called `curve_fit`

: `curve_fit(func, xdata, ydata, x0)`

. `curve_fit`

builds residuals function automatically and you can simply write:

```
import numpy as np
from scipy.optimize import curve_fit
x = np.asarray(range(20,401,20))
y = np.random.rand(20)
params = np.array([1,1])
def funcinv(x, a, b):
return b + a/x
res = curve_fit(funcinv, x, y, params)
print(res) # ... array([ 6.89518618, 0.37118815]), ...
```

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