import numpy as np
import matplotlib.pyplot as plt

t = np.arange(1,10)
y = np.random.uniform(0.0,3.0,size=9)
y = np.sort(y)

y0 = 2

# Adjusted Log Transformation (for when y is negative)
def transform(y):
    return np.log(1+y-np.min(y))

def invtransform(trans_y, orig_y):
    return np.exp(trans_y)-1+min(orig_y)

# Just some statistic
def compute_r(y0, ts, ys, n):
    r = 0
    for i in range(0,n):
        r = r + ts[i] * (np.log(ys[i])-np.log(y0)) / (ts[i] ** 2)
    return r

rs = []
for n in range(1,len(t)+1):
    rs.append(compute_r(y0, t, y, n))
rs = np.array(rs)

xs = np.arange(0, len(rs))
plt.plot(xs, np.array(rs), 'ro', label='orig.')
plt.plot(xs, transform(np.array(rs)), 'o', label='transformed')

transy = transform(np.array(rs))

coefs = np.polyfit(xs,transy, 1)
fit = coefs[1]+coefs[0]*xs
fitt = invtransform(fit, np.array(rs))

plt.plot(xs, fit, 'orange', label='fit', color='blue')
plt.plot(xs, fitt, 'orange', label='fit-backtransformed')
plt.legend(loc='upper right')
plt.show()