Can Box-Cox transformation be applied for data of this form?

Question

I have data of the form:

X       Y
3.53    0
4.93    50
5.53    60
6.21    70
7.37    80
9.98    90
16.56   100

And I want to find out $n$ so that this can be fit to a function of the form:

$$y = \frac{a}{x^n} + b$$

I am trying to determine $n$ by Box-Cox transformation. Can n be determined with Box-Cox transformation?

Answer 1

As they mentioned (and scolded you), why are you trying to use that method?

If a non-linear regression is enough you can see the code below, the option curve_fit did not work. So I used minimize in this case, the initial guess is obtained from a linear regression of $y$ and $(1/x)$.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize

def func(coef, x, y):
    a = coef[0]
    b = coef[1]
    n = coef[2]
    return np.linalg.norm(y - a/x**n - b)

x = np.array([3.53, 4.93, 5.53, 6.21, 7.37, 9.98, 16.56])
y = np.array([0, 50, 60, 70, 80, 90, 100])


# The initial values are obtained from a linear regression between
# y and (1/x)
res = minimize(func, [-446, 136, 1], args=(x, y))
a, b, n = res.x
print a, b, n
x_vec = np.linspace(3.5, 17)
y_vec = a/x_vec**n + b

plt.plot(x, y, 'ko')
plt.plot(x_vec, y_vec)
plt.xlabel("x")
plt.xlabel("y")
plt.savefig("regression.png", dpi=300)

The output is

-1181.72495119 105.101516419 1.91866624841

And the image