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Bootstrap Resampling

No, not Twitter Bootstrap - this bootstrapping is a way of sampling data, and it is one of the most important to consider what underlies... James Andrew Godwin February 2, 20218 min read

Simple, straightforward, convenient.

Alternate Data Dimension | Photo by Rene Böhmer Alternate Data Dimension | Photo by Rene Böhmer No, not Twitter Bootstrap - this bootstrapping is a way of sampling data, and it is one of the most important to consider what underlies the variation of numbers, the variation of distributions, what underlies distributions. To that end, bootstrapping works really, really well. For Data Scientists, Machine Learning Engineers, and statisticians alike it is vital to understand resampling methods. But why use resampling? We use resampling because we only have a limited amount of data - the limits of time, and economics, to say the least. What then is resampling? Resampling is when you take a sample, and then you take a sample of the sample. Why would you do that? Well, it allows you to see how much variation there would have been, it allows you to get a different understanding of the sample that you took. Say, for example, you wanted 1000 people to answer a survey, but you only have 100. By cleverly subsampling the samples we can get a new distribution, which we will know a little bit about the uncertainty of the sample; which we further presume is related to the uncertainty of the underlying population. We can compute statistics from multiple sub samples of data sets, such as the mean. We take a bunch of these items, we sum them up and divide them by the count by the number of items. The trick to bootstrap resampling is sampling with replacement. In Python, typically there will be a Boolean argument to your sampling parameter in your sampling code to your sampling function. This Boolean flag will be replace = true or replace = false. If you have 100 items, randomly sample (e.g., Bernoulli sample) the sample - record it, and replace that value back into the sample pool. Then rinse and repeat 99 more times. This subsample population is very likely to be different. They will have different proportions of the original 100 values. For the most part, the values are going to be pretty evenly distributed, but not perfectly. Then repeat this process a total of 10 times to generate 1000 samples. Note that occasionally some weird things can occur - e.g., if your data has one unique value of X - it is possible, but not likely due to sampling with replacement, that you can randomly subsample 100 values that include 5 X values, despite the sampled population only having one. For each of these samples, we can calculate the mean. And then we can look at the distribution of these means. Or we can look at the average of these means. The average of these means again should be pretty close to the population average, and should be pretty close to that larger sample average. You can do the same thing for other metrics, like standard deviations. Bootstrapping is a very powerful technique that allows you to get to another way of getting the uncertainties of your measurements. Because when you take those, when you get the means this variation of the means of all the sub samples will tell us quite a bit about the uncertainty that we would have that your sample has in respect to the true underlying population.

Bootstrap Benefits

Bootstrap and resampling are widely applicable statistical methods which relax many of the assumptions of classical statistics. Resampling methods implicitly draw on the Central Limit Theorem, as explained in my previous article. Specifically bootstrap (and other resampling methods):

Commonly used resampling methods include:

Bootstrap Overview

We compute the bootstrap mean as:

Bootstrap Example

To demonstrate the power of bootstrap I will analyze the means of the heights of different populations from Galton's height dataset in a Jupyter notebook; famous for giving us the phrase "regression to the mean" w/r to children's heights. The following is an example of a parametric bootstrap estimate - parametric because our model has a parameter, the mean, we are trying to estimate. I will demonstrate with an example: text

import pandas as pd
import numpy as np
from matplotlib import pyplot as plt
import seaborn as sns

text

%matplotlib inline

Importing the basic libraries text

# i imported the dataset from here, and called file_name
galton-families = pd.read_csv(file_name, index_col=0)
galton-families.head()

Check that the data is well-imported. Now I will subset the dataframe by gender to get the number of men and women in the dataset. This is going to create two data sets, one called male one called female. Male is actually sons and female is actually daughters. text

male = galton-families[galton-families.gender == 'male']
female = galton-families[galton-families.gender == 'female']

text

len(male), len(female)
(481, 453)

The set looks reasonably balanced with 481 males, 453 females in this dataset. The following code shows the min and max values in these distributions: text

print(families.childHeight.min())
families.childHeight.max()
56.0
Out[11]: 79.0

They range from 56 inches to 79 inches. The next thing to do is to plot these two histograms: text

def plot_hist(x, p=5):
 # Plot the distribution and mark the mean
 plt.hist(x, alpha=.5)
 plt.axvline(x.mean())
 # 95% confidence interval 
 plt.axvline(np.percentile(x, p/2.), color='red', linewidth=3)
 plt.axvline(np.percentile(x, 100-p/2.), color='red', linewidth=3)

def plot_dists(a, b, nbins, a_label='pop_A', b_label='pop_B', p=5):
 # Create a single sequence of bins to be shared across both
 # distribution plots for visualization consistency.
 combined = pd.concat([a, b])
 breaks = np.linspace(
 combined.min(), 
 combined.max(), 
 num=nbins+1)

text

 plt.subplot(2, 1, 1)
 plot_hist(a)
 plt.title(a_label)

 plt.subplot(2, 1, 2)
 plot_hist(b)
 plt.title(b_label)

 plt.tight_layout()

text

plot_dists(male.childHeight, female.childHeight, 20, a_label='sons', b_label='daughters')
plt.show()

The important thing to note here is that there is a significant overlap between sons and daughters. You can see that some of the daughters are actually taller than the mean of the sons. Overall, we can see that the daughters are smaller than the sons, but is it significant? My last post on the CLT demonstrated very clear ways of how to distinguish between these two distributions. It might look like these are two different distributions. However, we might actually say the distribution of the sons is not that different from the distributions of the daughters, because the 95% confidence level, which is indicated with these red vertical bars, sort of overlaps with the mean.

Bootstrap the Means

Bootstrap to the rescue. We are going to resample from our sample. Pandas has built-in support for generating bootstrap samples from a given dataframe. I will use the sample() method of our two dataframes to draw a single bootstrap sample like so: text

female.sample(frac=1, replace=True).head()

NB: With the replace equals true argument, you might get the same row multiple times, or some rows you are not going to sample at all. The above sample of all the daughters sampled from the Galton dataset has the following mean: text

female.sample(frac=1, replace=True).father.mean()
69.0664459161148

This bootstrapped sample of the female dataframe has a mean height of 69.1 inches for 453 daughters. Now we will take many (_n replicas) bootstrap samples and plot the distribution of sample means, as well as the mean of the sample, means. In the following code, we bootstrap 1000 subsamples each of the original size of the dataframe (481 and 453). text

n_replicas = 1000

text

female_bootstrap_means = pd.Series([
 female.sample(frac=1, replace=True).childHeight.mean()
 for i in range(n_replicas)])

text

male_bootstrap_means = pd.Series([
 male.sample(frac=1, replace=True).childHeight.mean()
 for i in range(n_replicas)])

text

plot_dists(male_bootstrap_means, female_bootstrap_means, 
 nbins=80, a_label='sons', b_label='daughters')
plt.show()

The distribution of the bootstrap means does not overlap at all! This proves that the difference is significant.

Bootstrap the Difference in Means

In the cell below, I will show how to generate bootstrap samples from the full male + female dataset and then difference in the means of male and female childHeight for each sample, generating a distribution of sample mean diffs. text

diffs = []
for i in range(n_replicas):
 sample = families.sample(frac=1.0, replace=True)
 male_sample_mean = sample[sample.gender == 'male'].childHeight.mean()
 female_sample_mean = sample[sample.gender == 'female'].childHeight.mean()
 diffs.append(male_sample_mean - female_sample_mean)
diffs = pd.Series(diffs)

text

plot_hist(diffs)

The distribution of the difference in means is far from zero. As like before, we can infer that the means of the two populations are significantly different - which for the purposes of this dataset is to say that sons and daughters do have different mean heights. Said differently: the confidence interval does not straddle zero, and because it does not straddle zero, we are confident that the difference is significant. The blue line indicates the mean difference between sons and daughters from the bootstrap sample of around 5.1 inches, of which we are 95% confident that the true population mean difference is between 4.8 inches and around 5.5 inches. This is the answer - that on average, sons are 5.5 inches taller than daughters. However, we must verify that the distribution of the difference in means normal, as implied by the CLT. If the data is not normally distributed then we cannot trust the 95% CI: text

import statsmodels.api as sm
fig = sm.qqplot(diffs, line='s')
plt.title('Quantiles of standard Normal vs. bookstrapped mean')
plt.show()

The points on the Q-Q Normal plot are nearly on a straight line. Apparently, the bootstrap distribution of the difference in means does conform to the CLT, which allows us to trust the statistics we derived from bootstrap resampling the original dataset.

Conclusion

The results of this analysis have demonstrated how useful a tool resampling is for a Data Scientist or Machine Learning Engineer. Bootstrap is not the only resampling method, there are several, but IMO it is the best for production models because it makes minimal assumptions about the parent distribution and is well implemented into many languages and packages. Data is useless if it is not properly treated and analyzed with the correct context. As Mark Twain famously said, "There are lies, damned lies, and statistics." Continue with me as I build on these concepts in Bayes statistics and using bootstrap with regression models. Find me on Linkedin Physicist cum Data Scientist- Available for new opportunity | SaaS | Sports | Start-ups | Scale-ups

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