Exercise - Univariate Gaussian Basics

Introduction

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 nn random and independant variables from a distribution (e.g. rolling a dice [1,6][1,6] 10 times, flipping a coin [0,1][0,1] 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 σ2\sigma^2 and
  • the mean μ\mu.

Remark: In order to detect errors in your own code, execute the notebook cells containing assert or assert_almost_equal. These statements raise exceptions, as long as the calculated result is not yet correct.

Requirements

Knowledge

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]

Python Modules

# External Modules
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

%matplotlib inline

Exercises

From an experiment we obtain a size NN random sample x1,,xN{\bf x}_1, \dots, {\bf x}_N from a Gaussian distribution:

P(xμ,σ)=12πσ2exp((xμ)22σ2)P(x\mid\mu,\sigma) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

with:

  • the mean μ\mu
  • the standard deviation σ\sigma
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 μ^\hat{\mu} for a data set D={x1,x2,xN}\mathcal D = \{x_1, x_2, \dots x_N \}:

μ^=1NxiNxi\hat{\mu} = \frac{1}{N} \sum_{x_i}^N x_i

with

-NN: Number of dat points

Task:

Implement the function to calculate the emprical mean without the use of the function np.mean.

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

Task:

Implement the function to calculate the sample variance with:

σ^N2=1Ni=1N(xiμ^)2\hat \sigma_{N}^2 = \frac{1}{N} \sum_{i=1}^N \left( x_i - \hat{\mu} \right)^2

resp.

σ^N12=1N1i=1N(xiμ^)2\hat \sigma_{N-1}^2 = \frac{1}{N-1} \sum_{i=1}^N \left( x_i - \hat{\mu} \right)^2

Your function should be able to handle both cases (and any NaN-a, a[0,N[a \in [0,N[), depending on the ddof argument (delta degrees of freedom). If the argument mean_ is 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))

Task:

  1. Sample mm such data sets and compute the estimator for the variance σ2\sigma^2 with ddof=0 and ddof=1:
  2. From the results of your simulation conclude which estimator σ^N\hat \sigma_N or σ^N1\hat \sigma_{N-1} 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))

Literature

Licenses

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.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.