Statistical Bootstrapping

• • Measure the margin of error of a small data sample.

This is about the data that we are using. Our Product Master is worried that we don’t have enough data to do anything meaningful. One way to solve this issue is to apply Statistical Bootstrapping or a type of bootstrapping called Case Resampling. We will apply this method to the problem of computing the mean of the AAPL stock price and the normal distribution.

The steps of the algorithm are:

• 1. Store the empirical distribution from our data.

  2. Generate random samples from this distribution of the same size as the original sample.

  3. Calculate and store the means of these samples.

  4. Determine in which percentile of the means distribution the mean of the original sample lies.

import numpy

import sys

import matplotlib.pyplot

from matplotlib.finance import quotes_historical_yahoo

from datetime import date

import scipy.stats

def random_indices(N):

   return numpy.random.randint(0, N, N)

def random_values(values):

   return numpy.take(values, random_indices(len(values)))

def generate_means(values):

   NTRIES = 400

   means = numpy.zeros(NTRIES)

   for i in xrange(NTRIES):

      means[i] = random_values(values).mean()

   return means

def format_mean(values):

   return "Mean=%.3f" % (values.mean())

def plot_percentile(values, means):

   matplotlib.pyplot.hist(means)

   percentile = scipy.stats.percentileofscore(means, values.mean())

   matplotlib.pyplot.legend([format_mean(means), "Percentile=%.2f" %(percentile)])

def plot(values):

   matplotlib.pyplot.hist(values)

   matplotlib.pyplot.legend([format_mean(values)])

today = date.today()

start = (today.year - 1, today.month, today.day)

quotes = quotes_historical_yahoo('AAPL', start, today)

close =  numpy.array([q[4] for q in quotes])

close_means = generate_means(close)

normal_values = numpy.random.normal(size=len(close))

normal_means = generate_means(normal_values)

matplotlib.pyplot.subplot(221)

matplotlib.pyplot.title("Close Values")

plot(close)

matplotlib.pyplot.subplot(222)

matplotlib.pyplot.title("Normal Values")

plot(normal_values)

matplotlib.pyplot.subplot(223)

matplotlib.pyplot.title("Close Means")

plot_percentile(close, close_means)

matplotlib.pyplot.subplot(224)

matplotlib.pyplot.title("Normal Means")

plot_percentile(normal_values, normal_means)

matplotlib.pyplot.show()