Probability plot correlation coefficient

The probability plot correlation coefficient (PPCC) is a graphical technique for identifying the shape parameter that best describes the dataset. Most of the statistical analysis has been done assuming the shape of the distribution in mind. However, these assumptions may be challenged because sometimes the distributions can have very different shapes depending upon the shape parameter. Therefore, it is better to find the shape parameter as part of the analysis, so that we can be more confident about the distribution of the population.

The PPCC plot is formed using the following axes:

• Vertical Axis: Probability plot correlation coefficient
• Horizontal Axis: Value of shape parameter

The main aim of the PPCC plot is first to find a good value of the shape parameter. In addition to calculating the shape parameter of the distribution, the PPCC plot can be used in deciding which distributional family is most appropriate.

The PPCC plot answers the following questions:

• What is the best-fit member within a distributional family?
• Does this best-fit member generate a good enough fit?
• Does this distributional family provide a good fit compared to other distributions?
• How sensitive is the choice of the shape parameter?

The Turkey-lambda PPCC plot, with shape parameter λ, is particularly useful for symmetric distributions. It indicates whether a distribution is short or long-tailed and it can further indicate several common distributions. Specifically,

• λ =-1, distribution is approximately Cauchy.
• λ = 0, distribution is exactly logistic.
• λ = 0.14, distribution is approximately normal.
• λ = 0.5, distribution is U-shaped.
• λ = 1, distribution is exactly uniform.

If the Turkey-Lambda PPCC plot gives a maximum value = 0.14, then we can conclude that the normal distribution is good approximate for the data. If the maximum value is < 0.14 then it means a long-tailed distribution such as the double exponential or logistic would be a better choice. If the maximum value is -1, then it implies a very-long tailed distribution such as Cauchy. If the maximum value is > 0.14 then it implies a very short-tailed distribution such as Beta or Uniform.

Implementation

• In this implementation, we will be generating different distribution and checking their Turkey-Lambda shape parameter value, and plotting PPCC plots. I am using Google Colaboratory, which contains some pre-installed libraries such as scipy, numpy, statsmodel, seaborn etc. However, these libraries can be easily installed using pip install in the local environment.

# import libraries
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as sc
import seaborn as sns

# generate different distributions
sample_size = 10000 
standard_norm = np.random.normal(size=sample_size)
cauchy_dist = sc.cauchy.rvs(loc=1, scale=10, size=sample_size)
logistic_dist = np.random.logistic(size=sample_size)
uniform_dist = np.random.uniform(size= sample_size)
beta_dist = np.random.beta(a=1, b=1, size=sample_size)

# Normal Distribution
fig, ax = plt.subplots(1, 2, figsize=(12, 7))
sns.histplot(standard_norm,kde=True, color ='blue',ax=ax[0])
sc.ppcc_plot(standard_norm, -5,5, plot=ax[1])
shape_param_normal = sc.ppcc_max(standard_norm)
ax[1].vlines(shape_param_normal,0,1, colors='red')
print("shape parameter of normal distribution is ", shape_param_normal)

# Cauchy Distribution
fig, ax = plt.subplots(1, 2, figsize=(12, 7))
sns.histplot(cauchy_dist, color ='blue',ax=ax[0])
ax[0].set_xlim(-40,40)
sc.ppcc_plot(cauchy_dist, -5,5, plot=ax[1])
shape_param_cauchy = sc.ppcc_max(cauchy_dist)
ax[1].vlines(shape_param_cauchy,0,1, colors='red')
print('shape parameter of cauchy distribution is ',shape_param_cauchy)

# Logistic Distribution
fig, ax = plt.subplots(1, 2, figsize=(12, 7))
sns.histplot(logistic_dist, color ='blue',ax=ax[0])
sc.ppcc_plot(logistic_dist, -5,5, plot=ax[1])
shape_param_logistic = sc.ppcc_max(logistic_dist)
ax[1].vlines(shape_param_logistic,0,1, colors='red')
print("shape parameter of logistic is ",shape_param_logistic)

# Uniform Distribution
fig, ax = plt.subplots(1, 2, figsize=(12, 7))
sns.histplot(uniform_dist, color ='green',ax=ax[0])
sc.ppcc_plot(uniform_dist, -5,5, plot=ax[1])
shape_para_uniform =sc.ppcc_max(uniform_dist)
ax[1].vlines(shape_para_uniform,0,1, colors='red')
print("shape parameter of uniform distribution is ",shape_para_uniform)

# Beta Distribution
fig, ax = plt.subplots(1, 2, figsize=(12, 7))
sns.histplot(beta_dist, color ='blue',ax=ax[0])
sc.ppcc_plot(beta_dist, -5,5, plot=ax[1])
shape_para_beta =sc.ppcc_max(beta_dist)
ax[1].vlines(shape_para_beta,0,1, colors='red')
print("shape parameter of beta distribution is :",shape_para_beta)

shape parameter of normal distribution is  0.14139046072745928

shape parameter of cauchy distribution is  -0.8555566289941865

shape parameter of logistic is  0.003792036190661425

shape parameter of uniform distribution is  1.0681942803525217

shape parameter of beta distribution is : 0.9158983492057267

References

• PPCC plot NIST