Generate random points above and below a line in Python

Question

I would like to generate random points on an x,y scatter plot that are either above or below a given line. For example, if the line is y=x I would like to generate a list of points in the top left of the plot (above the line) and a list of points in the bottom right of the plot (below the line). Here's is an example where the points are above or below y=5:

import random
import matplotlib.pyplot as plt

num_points = 10
x1 = [random.randrange(start=1, stop=9) for i in range(num_points)]
x2 = [random.randrange(start=1, stop=9) for i in range(num_points)]
y1 = [random.randrange(start=1, stop=5) for i in range(num_points)]
y2 = [random.randrange(start=6, stop=9) for i in range(num_points)]

plt.scatter(x1, y1, c='blue')
plt.scatter(x2, y2, c='red')
plt.show()

However, I generated the x and y points independently, which limits me to equations where y = c (where c is a constant). How can I expand this to any y=mx+b?

Answer 1

You can change the stop and start limits for y1 and y2 to be the line you want. You will need to decide where the plane ends (set lower and upper).

Note this only works for integers. You can use truncated multivariate distributions if you want something more sophisticated.

m, b = 1, 0
lower, upper = -25, 25

x1 = [random.randrange(start=1, stop=9) for i in range(num_points)]
x2 = [random.randrange(start=1, stop=9) for i in range(num_points)]

y1 = [random.randrange(start=lower, stop=m*x+b) for x in x1]
y2 = [random.randrange(start=m*x+b, stop=upper) for x in x2]

plt.plot(np.arange(10), m*np.arange(10)+b)
plt.scatter(x1, y1, c='blue')
plt.scatter(x2, y2, c='red')

Answer 2

You may as well have my answer too.

This way puts Gaussian noise above the line, and below. I have deliberately set the mean of the noise to 20 so that it would stand out from the line, which is y = 10*x + 5. You would probably make the mean zero.

>>> import random
>>> def y(x, m, b):
...     return m*x + b
... 
>>> import numpy as np
>>> X = np.linspace(0, 10, 100)
>>> y_above = [y(x, 10, 5) + abs(random.gauss(20,5)) for x in X]
>>> y_below = [y(x, 10, 5) - abs(random.gauss(20,5)) for x in X]
>>> import matplotlib.pyplot as plt
>>> plt.scatter(X, y_below, c='g')
>>> plt.scatter(X, y_above, c='r')
>>> plt.show()

Here's the plot.

Answer 3

There are many approaches possible, but if your only requirement is that they are above and below the y = mx + b line, then you can simply plug the random x values into the equation and then add or subtract a random y value.

import random
import matplotlib.pyplot as plt

slope = 1
intercept = 0

def ymxb(slope, intercept, x):
    return slope * x + intercept

num_points = 10
x1 = [random.randrange(start=1, stop=9) for i in range(num_points)]
x2 = [random.randrange(start=1, stop=9) for i in range(num_points)]
y1 = [ymxb(slope, intercept, x) - random.randrange(start=1, stop=9) for x in x1]
y2 = [ymxb(slope, intercept, x) + random.randrange(start=1, stop=9) for x in x2]

plt.scatter(x1, y1, c='blue')
plt.scatter(x2, y2, c='red')
plt.show()

That looks like this: