Bootstrapping in Python

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Introduction

The term "bootstrap" is euphonic with the metaphor of pulling oneself up by one's bootstraps, signifying the achievement of the seemingly impossible without external assistance. Analogously, as seen in Raspe's tales of Baron Munchausen, who extricated himself and his horse from a swamp by his own hair, the Bootstrap technique enables the attainment of statistical insights independently, bypassing the need for predefined formulas.

Bootstrap is applicable for any samples. It is useful when:

1. observations are not described by normal distribution;
2. there are no statistical tests for the desired quantities.

Theoretical aspect

Bootstrapping (sometimes also just bootstrap) is a method for estimating statistical quantities without reliance on explicit formulas.

To derive a specified statistic for a sample (such as the mean or variance, quantile, etc.), using bootstrap, pseudo-samples or subsamples are generated from the original sample by drawing with replacement. The mean (variance, quantile, etc.) is then computed for each pseudo-sample. Theoretically, this process can be repeated multiple times, generating numerous instances of the parameter of interest and allowing for the assessment of its distribution.

Figure 1. Illustration of bootstrapping method. Source: Wikipedia

Advantages:

1. Simplicity and flexibility.
2. No knowledge about data distribution is needed.
3. Helps to avoids the cost of repeating the experiment to get more data.
4. Usually provides a more accurate estimates of a statistic than "standard" methods utilizing normal distribution.

Disadvantages:

1. Computational intensity. The process involves resampling from the dataset multiple times, which might be time-consuming for extensive datasets.
2. Dependence on the original data. Bootstrapping assumes that the original dataset is representative of the population.
3. Sensitivity to outliers. Since bootstrap samples with replacement, outliers may be overrepresented in the resampled datasets, affecting the stability of the results.

Synthetic examples of bootstrapping

Example 1. Bootstrapping for quantile evaluation

We will generate a sample with some random distribution:

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

np.random.seed(42)

data = pd.Series(np.random.rand(100) * 100)

data.hist(figsize=(5, 3))
plt.title("Example 1: initial data histogram")
plt.show()

Using bootstrap to evaluate outliers in this case is much more reliable, than calculating normal distribution quantiles. For this purpose we will generate 1000 subsamples using pandas sample function and a loop. It is important to set replace=True, as bootstrap relies on drawing samples with replacement. We will evaluate the upper 0.99 and lower 0.01 quantiles.

upper_quantiles = []
lower_quantiles = []
for i in range(1000):
    subsample = data.sample(frac=1, replace=True)
    upper_quantiles.append(subsample.quantile(0.99))
    lower_quantiles.append(subsample.quantile(0.01))

upper_quantiles = pd.Series(upper_quantiles)
lower_quantiles = pd.Series(lower_quantiles)

We get an experimental distribution for both quantiles:

upper_quantiles.hist(figsize=(5, 3))
plt.title("Example 1: upper 0.99 quantile bootstrap simulation")
plt.show()

Now we can estimate confidence intervals for the upper and lower 0.01 quantiles:

# upper 0.99 quantile lies in this interval with 95% confidence
print(f'{upper_quantiles.quantile(0.025):.2f}') # Out: 94.89
print(f'{upper_quantiles.quantile(0.975):.2f}') # Out: 98.69

# lower 0.01 quantile lies in this interval with 95% confidence
print(f'{lower_quantiles.quantile(0.025):.2f}') # Out: 0.55
print(f'{lower_quantiles.quantile(0.975):.2f}') # Out: 4.62

Example 2. Bootstrapping as an alternative to t-test

Bootstrap can be implemented for analyzing difference between mean values of two samples.

# generate 2 samples
samples_A = pd.Series(
    [100.24,  97.77,  95.56,  99.49, 101.4 , 105.35,  93.33,  93.02,
     101.37,  95.66,  93.34, 100.75, 104.93,  97.  ,  95.46, 100.03,
     102.34,  93.23,  97.05,  97.76,  93.63, 100.32,  99.51,  99.31,
     102.41, 100.69,  99.67, 100.99],
    name='sample_A')
samples_B = pd.Series(
    [101.67, 102.27,  97.01, 103.46, 100.76, 101.19,  99.11,  97.59,
     101.01, 101.45,  94.3 , 101.55,  96.33,  99.03, 102.33,  97.32,
      93.25,  97.17, 101.1 , 102.57, 104.59, 105.63,  93.93, 103.37,
     101.62, 100.62, 102.79, 104.19],
    name='sample_B')

plt.show()

AB_difference = samples_B.mean() - samples_A.mean()
print(f"Difference of mean values {AB_difference:.2f}") # Out: Difference of mean values 1.63

Let us calculate the probability that we observe such difference (1.63 in this case) only by chance and how close the means are. We will run bootstrap simulation as follows:

1. Randomly sample with replacement from each group separately. The size of samples is equal to the size of the original group.
2. Calculate the difference of means between two created samples.
3. Repeat steps 1-2 n times. (We will choose n as num_samples = 1000).
4. Count p-value, as a proportion of bootstrap samples where the absolute value of mean difference is greater than or equal to the absolute value of the observed mean difference.

alpha = 0.05 # significance level 
num_samples = 1000
count = 0
diffs = []

for i in range(num_samples):
    subsample_A = samples_A.sample(n=len(samples_A), replace=True)
    subsample_B = samples_B.sample(n=len(samples_B), replace=True)

    # calculate difference of means of 2 subsamples
    bootstrap_difference = subsample_B.mean() - subsample_A.mean()
    diffs.append(bootstrap_difference)

for el in diffs:
    if abs(el) >= abs(AB_difference):
        count+=1

pvalue = count / num_samples
print('p-value =', pvalue)

if pvalue < alpha:
    print("Reject H0: mean values do differ significantly")
else:
    print("Accept H0: mean values do not differ significantly")

p-value = 0.514
Accept H0: mean values do not differ significantly

Visualize the resulted distribution of saved mean differences

pd.Series(diffs).hist(bins=20, figsize=(5, 4))
plt.title(
    "Example 2: bootstrap simulation. The difference of mean values of 2 samples")
plt.show()

This histogram represents the variability of the mean difference that could arise from sampling variations. From the figure we conclude that value of 1.63 is not occurred by chance.

t-test

ttest_ind(samples_A.values, samples_B.values)

TtestResult(statistic=-1.7971305016442471, pvalue=0.07790788332728987, df=54.0)Out: TtestResult(statistic=-1.7971305016442471, pvalue=0.07790788332728987, df=54.0)

p-value of t-test is a bit higher than p-value of the bootstrap sampling, but they are both more than significance level. Therefore, we can say, that the mean values do not differ significantly. From the histigram we also could observe the values fluctuate from -2 to 5, having in mind that the mean values of samples A and B are 98.63 and 100.26, respectively.