A bootstrap distribution is the frequency distribution of a statistic calculated across many datasets resampled from the original data. Each resample matches the original size and selects observations with replacement, so values may repeat or be omitted. section 3.4.4
Use the distribution to estimate a population parameter’s confidence interval:
- Resample the original observations many times with replacement. section 3.4.4
- Calculate a statistic, such as a mean, median, or regression coefficient, for every resample. section 3.4.4
- Use the resulting statistic distribution to estimate a confidence interval, especially when the underlying distribution is unknown or difficult to determine analytically. section 3.4.4
Estimating equipment safety margins
After 50 destructive tests produce breakpoint temperatures, repeatedly resample 50 temperatures with replacement—perhaps 10,000 times—and examine the resulting distribution instead of relying only on the average. section 3.4.4
Listing 3.8 Calculating confidence intervals for sales and review scores by state
# Prepare datasets
# Merge df_products with df_product_category_translation
products_merged = pd.merge(
df_products, df_product_category_translation, on='product_category_name')
# Merge df_orders with df_order_items
order_items_merged = pd.merge(df_orders, df_order_items, on='order_id')
# Merge the resulting DataFrame with df_customers
order_customer_merged = pd.merge(
order_items_merged, df_customers, on='customer_id')
# Merge the resulting DataFrame with products_merged
order_product_merged = pd.merge(
order_customer_merged, products_merged, on='product_id')
# Merge the resulting DataFrame with df_sellers
order_seller_merged = pd.merge(
order_product_merged, df_sellers, on='seller_id')
# Merge the resulting DataFrame with df_order_reviews
order_review_merged = pd.merge(
order_seller_merged, df_order_reviews, on='order_id')
# Merge the resulting DataFrame with df_order_payments
analysis_df = pd.merge(
order_review_merged, df_order_payments, on='order_id')
# Choose a specific product category to analyze
product_category = "health_beauty"
filtered_df = analysis_df[
analysis_df['product_category_name_english'] == product_category]
# Group the data by seller state or city, and calculate
the metrics of interest
grouped_data = filtered_df.groupby('seller_state')
group_metrics = grouped_data.agg(
{'price': ['sum', 'mean', 'count'], 'review_score': 'mean'})
# Calculate the standard deviations
group_metrics['price', 'std'] = grouped_data['price'].std()
group_metrics['review_score', 'std'] = grouped_data['review_score'].std()
# Use bootstrapping to estimate the 95% confidence intervals for
the average sales revenue and average review score for each city or state.
import numpy as np
# Bootstrap the confidence intervals
def bootstrap_CI(data, func, n_bootstraps=1000, ci=95, axis=0):
"""
Generate a confidence interval for a given statistic using
bootstrapping.
Parameters:
data (numpy.ndarray or pandas.DataFrame):
The data to calculate the statistic from.
func (callable): The function used to calculate the statistic.
n_bootstraps (int): The number of bootstrap samples to generate.
ci (float): The confidence interval percentage (e.g., 95 for a 95% CI).
axis (int):
The axis along which to apply the statistic function
(0 for columns, 1 for rows).
Returns:
tuple: The lower and upper bounds of the confidence interval.
"""
bootstrapped_statistics = []
for _ in range(n_bootstraps):
bootstrap_sample = np.random.choice(
data, size=len(data), replace=True)
bootstrapped_statistic = func(bootstrap_sample, axis=axis)
bootstrapped_statistics.append(bootstrapped_statistic)
lower_bound = np.percentile(bootstrapped_statistics, (100 - ci) / 2)
upper_bound = np.percentile(
bootstrapped_statistics, 100 - (100 - ci) / 2)
return lower_bound, upper_bound
# Create a new DataFrame to store the confidence intervals
def calculate_ci(group):
return pd.Series({
'price_ci_lower': bootstrap_CI(group['price'], np.mean,
n_bootstraps=1000, ci=95)[0],
'price_ci_upper': bootstrap_CI(group['price'], np.mean,
n_bootstraps=1000, ci=95)[1],
'review_score_ci_lower': bootstrap_CI(group['review_score'],
np.mean, n_bootstraps=1000, ci=95)[0],
'review_score_ci_upper': bootstrap_CI(group['review_score'],
np.mean, n_bootstraps=1000, ci=95)[1]
})
ci_df = grouped_data.apply(calculate_ci)
# Create visualizations with error bars for the average sales revenue and
average review score by seller state.
def plot_error_bars_only(df, metric_name, ylabel='', title='',
figsize=(10, 6)):
fig, ax = plt.subplots(figsize=figsize)
x = np.arange(len(df.index))
y = df[metric_name + '_ci_lower'] + (df[metric_name + '_ci_upper'] –
df[metric_name + '_ci_lower']) / 2
yerr = (df[metric_name + '_ci_upper'] –
df[metric_name + '_ci_lower']) / 2
plt.errorbar(x, y, yerr=yerr, fmt='o', capsize=5, capthick=2,
ecolor='black', elinewidth=2)
plt.xticks(x, df.index, rotation=90)
plt.xlabel('Seller State')
plt.ylabel(ylabel)
plt.title(title)
plt.grid(axis='y')
plt.show()
# Plot the confidence intervals
plot_error_bars_only(ci_df, 'price', ylabel='Average Sales')
plot_error_bars_only(ci_df, 'review_score', ylabel='Average Review Score')
Inspect a Python workflow that calculates confidence intervals for sales prices and review scores by state. section 3.4.4
