Exercise - Univariate Gaussian Basics
Table of Contents
The normal distribution, also Gaussian distribution, is a distribution, which you can encounter endless times in a lot of different domains. This is because of the central limit theorem (CLT): When you draw random and independant variables from a distribution (e.g. rolling a dice 10 times, flipping a coin 10 times, etc...) and you calculate the mean or the sum of your sample, then the mean (or sum) will converge to a Gaussian distribution if you repeat this process several times.
Furthermore, the Gaussian has a very convenient PDF since we only need two parameters to describe it:
- the variance and
- the mean .
Remark: In order to detect errors in your own code, execute the notebook cells containing
assert_almost_equal. These statements raise exceptions, as long as the calculated result is not yet correct.
To complete this exercise notebook, you should possess knowledge about the following topics.
- Univariate Gaussian
- Empirical mean
- Variance / sample variance The following material can help you to acquire this knowledge:
- Gaussian, variance, mean:
- Chapter 3 of the Deep Learning Book [GOO16]
- Chapter 1 of the book Pattern Recognition and Machine Learning by Christopher M. Bishop [BIS07]
- Univariate gaussian:
- Video1 and the follwoing of Khan Academy [KHA18a]
- Sample variance:
- Video2 and the follwoing of Khan Academy [KHA18b]
# External Modules import numpy as np import matplotlib.pyplot as plt import scipy.stats as stats %matplotlib inline
From an experiment we obtain a size random sample from a Gaussian distribution:
- the mean
- the standard deviation
mu = -1.5 sigma = 3 sigma_square = sigma**2 size = 10
def plot_gaussian_pdf(mu, sigma): x = np.linspace(mu - 4*sigma, mu + 4*sigma, 100) plt.plot(x,stats.norm.pdf(x, mu, sigma)) plot_gaussian_pdf(mu, sigma)
def get_data(mu, sigma_square, size): sigma = np.sqrt(sigma_square) x = np.random.normal(loc=mu, scale=sigma, size=size) return x
x = get_data(mu, sigma_square, size) x
def plot_hist(x): fig = plt.figure(figsize=(12,5), dpi=80) ax = fig.add_subplot(121) ax.hist(x, bins=3) ax.set_xlabel(r'$x$', fontsize=20) ax.set_ylabel(r'$c$', fontsize=20) ax.set_title("Histogram of sample x", fontsize=20) plot_hist(x)
Exercise - Empirical Mean
To calculate the empricial mean for a data set :
-: Number of dat points
Implement the function to calculate the emprical mean without the use of the function
def mean(x): """ Calculates the mean of x """ raise NotImplementedError
np.testing.assert_almost_equal(mean(x), x.mean()) print(mean(x))
Exercise - Sample Variance
Implement the function to calculate the sample variance with:
Your function should be able to handle both cases (and any , ), depending on the
ddof argument (delta degrees of freedom). If the argument
None use the empirical mean.
def var(x, mean_=None, ddof=0): """ Calculates the variance of x :x: sample :x type: numpy array type float :mean_: mean to use for the calculation. if mean=None, the empirical mean of x will be used :mean_ type: float or None :ddof: delat degrees of freedom :ddof type: integer :return: the variance of x :r type: float """ raise NotImplementedError
np.testing.assert_almost_equal(var(x, ddof=1), np.var(x, ddof=1)) np.testing.assert_almost_equal(var(x, ddof=0), np.var(x, ddof=0))
- Sample such data sets and compute the estimator for the variance with
- From the results of your simulation conclude which estimator or is a biased resp. unbiased estimator?
def get_sigma_square_estimate(m, mu, sigma_square, size, ddof=0): """ Estimates the variance of m Gaussian samples using their empirical variance :m: number of samples :m type: integer :mu: mean of the Gaussian :mu type: float :size: size of each sample :size type: unsigned integer :sigma_square: sigma_square (variance) of the gaussian :sigma_square type: float :ddof: delat degrees of freedom :ddof type: integer :return: estimated variance :r type: float """ raise NotImplementedError
m = 100000 print("ddof=0: var:\t", get_sigma_square_estimate(m, mu, sigma_square, size, ddof=0)) print("ddof=1: var:\t", get_sigma_square_estimate(m, mu, sigma_square, size, ddof=1))
Notebook License (CC-BY-SA 4.0)
The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).
Exercise - Multivariate Gaussian
by Christian Herta, Klaus Strohmenger
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.
Code License (MIT)
The following license only applies to code cells of the notebook.
Copyright 2018 Christian Herta, Klaus Strohmenger
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
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