Central Limit Theorem In Machine Learning

Central Limit theorem says:

Implementing Central Limit Theorm Using BlackFriday.csv Dataset

Import Libraries

import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import random
random.seed = 42
import warnings
warnings.filterwarnings("ignore")

import plotly.offline as offline
import plotly.graph_objs as go
offline.init_notebook_mode()
import pandas as pd
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from prettytable import PrettyTable
from IPython.display import HTML, display
plt.xkcd()

Read Dataset

# BlackFriday.csv
# The dataset here is a sample of the transactions made in a retail store on BlackFriday
# https://www.kaggle.com/mehdidag/black-friday

df = pd.read_csv('BlackFriday.csv')
print("number of data points in our popultion:", df.shape)
print("% of missing values",df['Purchase'].isnull().sum() * 100 / len(df))
df.head(2)

Output:

number of data points in our popultion: (537577, 12) % of missing values 0.0

Find Mean and Standard Deviation of Population

data = np.array(df['Purchase'].values)
print("Number of samples in our data: ",data.shape[0])
sns.distplot(data, color='g')
plt.show()

# population mean
population_mean = np.round(data.mean(),3)

# population std
population_std = np.round(data.std(),3)

Output

Number of samples in our data: 537577

Now Let's take 100 samples with each of size 100, and try to plot the distribution of their 'mean'

def get_means_of_n_samples_with_m_size(data, n, m):
    sample_mean_m_samples_n_ele = []
    for i in range(0,n):
        samples = random.sample(range(0, data.shape[0]), m)
        sample_mean_m_samples_n_ele.append(data[samples].mean())
    return sample_mean_m_samples_n_ele

Central Limit Theorm

def central_limit_theorem(data, population_mean , i, j, color, key):
    sns.distplot(np.array(data), color=color, ax=axs[i, j])
    axs[i, j].axvline(population_mean, linestyle="--", color='r', label="p_mean")
    axs[i, j].axvline(np.array(data).mean(), linestyle="-.", color='b', label="s_mean")
    axs[i, j].set_title(key)
    axs[i, j].legend()

sample_means = dict()

sample_means['100samples_50ele'] = get_means_of_n_samples_with_m_size(data,100, 50)
sample_means['1000samples_50ele'] = get_means_of_n_samples_with_m_size(data,1000, 50)

sample_means['100samples_100ele'] = get_means_of_n_samples_with_m_size(data,100, 100)
sample_means['1000samples_100ele'] = get_means_of_n_samples_with_m_size(data,1000, 100)

sample_means['100samples_1000ele'] = get_means_of_n_samples_with_m_size(data,100, 1000)
sample_means['1000samples_1000ele'] = get_means_of_n_samples_with_m_size(data,1000, 1000)

Now Let's take 1000 samples with each of size 100, and try to plot the distribution of their 'mean'

#red, green, blue, yellow, etc
colrs = ['r','g','b','y', 'c', 'm', 'k']
plt_grid  = [(0,0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]
sample_sizes = [(100,50), (1000, 50), (100, 100), (1000, 100), (100, 1000), (100, 1000)]

fig, axs = plt.subplots(3, 2,  figsize=(10, 10))
for i, key in enumerate(sample_means.keys()):
    central_limit_theorem(sample_means[key], population_mean , plt_grid[i][0], plt_grid[i][1] , colrs[i], key)
plt.show()

Output

if we can observe the thrid row distribution plots, the larger the sample size, the more it looks like Gaussian

Note : For the Central limit theorem to be valid, the samples have to be reasonably large. How large is that? It depends on how far the population distribution differs from a Gaussian distribution. Assuming the population doesn't have a really unusual distribution, a sample size of 10 or so is generally enough to invoke the Central Limit Theorem.

__ Let us get the properties of these sample distributions and compare these stats with the original distribution__

x = PrettyTable()
x = PrettyTable(["#samples_name", "P_Mean", "Sampel mean", "P_Std", "Sample Std", "mu_x"+u"\u2248"+"mu", "std_x"+u"\u2248"+"std/"+u"\u221A"+"n"])

for i, key in enumerate(sample_means.keys()):
    sample_mean = np.round(np.array(sample_means[key]).mean(), 3)
    sample_std = np.round(np.array(sample_means[key]).std(), 3)
    population_std_est = np.round(population_std/np.sqrt(sample_sizes[i][1]), 3)
    
    row = []
    row.append(key)
    row.append(population_mean)
    row.append(sample_mean)
    row.append(population_std)
    row.append(sample_std)
    row.append(str(sample_mean)+u"\u2248"+str(population_mean))
    row.append(str(sample_std)+u"\u2248"+str(population_std_est))
    x.add_row(row)
print(x)

Output

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