How to plot upper and lower boundary with a LINEAR line on a scatter plot?

Question

I have a data frame df with columns A and Q. I am using this code to draw a line of equation on it.

#Actual line of equation, which has to be plotted: Q=alpha*A^beta : ln(Q)=a+b*ln(A) : y = a+b(x)

x = np.log(df['A'])
y = np.log(df['Q'])

#deriving b,a
b,a = np.polyfit(np.log(x), y, 1)

#deriving alpha and beta. By using a = ln(alpha); b = beta -1
alpha = np.exp(a)
beta = b + 1

Q = df['Q'].values
A = df['A'].values

#equation of line
q = alpha * np.power(A,beta)

#plotting the points and line
plt.scatter(A,Q)
plt.plot(A,q, '-r')
plt.yscale('log')
plt.xscale('log')

This gives the following output, which is similar to a regression line.

But I am interested in plotting the same line of the equation as the upper and lower curve/boundary joining the farthest points(perpendicular to the green line) on both sides as shown below with the same slope as that of the continuous green line.

Answer 1

The idea is to first search the index of the point where the difference between the line and the plot is minimal (cf. maximal). With this point, alpha_min can be calculated such that

Q[pos_min] == alpha_min * np.power(A[pos_min], beta), thus

alpha_min = Q[pos_min] / np.power(A[pos_min], beta).

As such lines can extend quite far away from the original points, it can help to restore the x and y limits (thus clipping the plot to the original region).

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

df = pd.DataFrame()
df['A'] = 10 ** np.random.uniform(0, 1, 1000) ** 2
df['Q'] = 10 ** np.random.uniform(0, 1, 1000) ** 2

x = np.log(df['A'])
y = np.log(df['Q'])

# deriving b,a
b, a = np.polyfit(np.log(x), y, 1)

# deriving alpha and beta. By using a = ln(alpha); b = beta - 1
alpha = np.exp(a)
beta = b + 1

Q = df['Q'].values
A = df['A'].values

# plotting the points and line
plt.yscale('log')
plt.xscale('log')
plt.scatter(A, Q, color='b')

# equation of line
xmin, xmax = plt.xlim() # the limits of the x-axis for drawing the line
x = np.linspace(xmin, xmax, 50)
q = alpha * np.power(x, beta)
plt.plot(x, q, '-r')
ymin, ymax = plt.ylim()  # store the limits of the scatter and line plot so they can be restored later

pos_min = np.argmin(Q / np.power(A, beta))
pos_max = np.argmax(Q / np.power(A, beta))

alpha_min = Q[pos_min] / np.power(A[pos_min], beta)
alpha_max = Q[pos_max] / np.power(A[pos_max], beta)

# plt.scatter(A[pos_min], Q[pos_min], s=100, fc='none', ec='r', lw=3)
# plt.scatter(A[pos_max], Q[pos_max], s=100, fc='none', ec='g', lw=3)

plt.plot(x, (alpha_max) * np.power(x, beta), '--r')
plt.plot(x, (alpha_min) * np.power(x, beta), '--r')

plt.xlim(xmin, xmax)  # restore the limits of the scatter plot
plt.ylim(ymin, ymax)
plt.show()