How to random sample lognormal data in Python using the inverse CDF and specify target percentiles?

Question

I'm trying to generate random samples from a lognormal distribution in Python, the application is for simulating network traffic. I'd like to generate samples such that:

1. The modal sample result is 320 (~10^2.5)
2. 80% of the samples lie within the range 100 to 1000 (10^2 to 10^3)

My strategy is to use the inverse CDF (or Smirnov transform I believe):

1. Use the PDF for a normal distribution centred around 2.5 to calculate the PDF for 10^x where x ~ N(2.5,sigma).
2. Calculate the CDF for the above distribution.
3. Generate random uniform data along the interval 0 to 1.
4. Use the inverse CDF to transform the random uniform data into the required range.

The problem is, when I calculate the 10 and 90th percentile at the end, I have completely the wrong numbers.

Here is my code:

%matplotlib inline

import matplotlib
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats
from scipy.stats import norm

# find value of mu and sigma so that 80% of data lies within range 2 to 3
mu=2.505
sigma = 1/2.505
norm.ppf(0.1, loc=mu,scale=sigma),norm.ppf(0.9, loc=mu,scale=sigma)
# output: (1.9934025, 3.01659743)

# Generate normal distribution PDF
x = np.arange(16,128000, 16) # linearly spaced here, with extra range so that CDF is correctly scaled
x_log = np.log10(x)
mu=2.505
sigma = 1/2.505
y = norm.pdf(x_log,loc=mu,scale=sigma)
fig, ax = plt.subplots()
ax.plot(x_log, y, 'r-', lw=5, alpha=0.6, label='norm pdf')

x2 = (10**x_log) # x2 should be linearly spaced, so that cumsum works (later)
fig, ax = plt.subplots()
ax.plot(x2, y, 'r-', lw=5, alpha=0.6, label='norm pdf')
ax.set_xlim(0,2000)

# Calculate CDF
y_CDF = np.cumsum(y) / np.cumsum(y).max()
fig, ax = plt.subplots()
ax.plot(x2, y_CDF, 'r-', lw=2, alpha=0.6, label='norm pdf')
ax.set_xlim(0,8000)

# Generate random uniform data
input = np.random.uniform(size=10000)

# Use CDF as lookup table
traffic = x2[np.abs(np.subtract.outer(y_CDF, input)).argmin(0)]

# Discard highs and lows
traffic = traffic[(traffic >= 32) & (traffic <= 8000)]

# Check percentiles
np.percentile(traffic,10),np.percentile(traffic,90)

Which produces the output:

(223.99999999999997, 2480.0000000000009)

... and not the (100, 1000) that I would like to see. Any advice appreciated!

Answer 1

First, I'm not sure about Use the PDF for a normal distribution centred around 2.5. After all, log-normal is about base e logarithm (aka natural log), which means 320 = 10^2.5 = e^5.77.

Second, I would approach problem in a different way. You need m and s to sample from Log-Normal.

If you look at wiki article above, you could see that it is two-parametric distribution. And you have exactly two conditions:

Mode = exp(m - s*s) = 320
80% samples in [100,1000] => CDF(1000,m,s) - CDF(100,m,s) = 0.8

where CDF is expressed via error function (which is pretty much common function found in any library)

So two non-linear equations for two parameters. Solve them, find m and s and put it into any standard log-normal sampling