Tag: rand function

Non-central Chi square distribution

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If squares of k independent standard normal random variables are added, it gives rise to central Chi-squared distribution with ‘k’ degrees of freedom. Instead, if squares of k independent normal random variables with non-zero means are added, it gives rise to non-central Chi-squared distribution. Non-central Chi-square distribution is related to Ricean distribution, whereas the central Chi-squared distribution is related to Rayleigh distribution.

The non-central Chi-squared distribution is a generalization of Chi-square distribution. A non-central Chi squared distribution is defined by two parameters: 1) degrees of freedom () and 2) non-centrality parameter .

As we know from previous article, the degrees of freedom specify the number of independent random variables we want to square and sum-up to make the Chi-squared distribution. Non-centrality parameter is the sum of squares of means of the each independent underlying normal random variable.

The non-centrality parameter is given by

The PDF of the non-central Chi-squared distribution having degrees of freedom and non-centrality parameter is given by

Here, the random variable is central Chi-squared distributed with degrees of freedom. The factor gives the probabilities of Poisson distribution. Thus, the PDF of the non-central Chi-squared distribution can be termed as the weighted sum of Chi-squared probabilities where the weights being equal to the probabilities of Poisson distribution.

Method of Generating non-central Chi-squared random variable:

The procedure for generating the samples from a non-central Chi-squared random variable is as follows.

● For a given degree of freedom , let the normal random variables be with variances and mean respectively.
● The goal is to add squares of these independent normal random variables with variances set to one and means satisfying the condition set by equation (1).
● Set and
● Generate standard normal random variables and one normal random variable with and
● Squaring and summing-up all the random variables gives the non-central Chi-squared random variable.
● The PDF of the generated samples can be plotted using the histogram method described here.

Matlab Code:

Check this book for full Matlab code.
Wireless Communication Systems using Matlab – by Mathuranathan Viswanathan

Python Code:

Python numpy package has a nocentral_chisquare() generator, which can be used in a straightforward manner to obtain the non-central Chi square distributed sequences.

#---------Non-central Chi square distribution gaussianwaves.com-----
import numpy as np
import matplotlib.pyplot as plt
#%matplotlib inline
plt.style.use('ggplot')

ks=np.asarray([2,4]) #degrees of freedoms to simulate
ldas = np.asarray([1,2,3]) #non-centrality parameters to simulate
nSamp=1000000 #number of samples to generate

fig, ax = plt.subplots(ncols=1, nrows=1, constrained_layout=True)

for i,k in enumerate(ks):
    for j,lda in enumerate(ldas):
        #Generate non-central Chi-squared distributed random numbers
        X = np.random.noncentral_chisquare(df=k, nonc = lda, size = nSamp)
        ax.hist(X,bins=500,density=True,label=r'$k$={} $\lambda$={}'.format(k,lda),\
        histtype='step',alpha=0.75, linewidth=3)

ax.set_xlim(left=0,right=30);ax.legend()
ax.set_title('PDFs of non-central Chi square distribution');
plt.show()
Simulated PDFs of non-central Chi-Squared random variables
Figure 1: Simulated PDFs of non-central Chi-Squared random variables

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For further reading

[1] David A. Harville, “Linear Models and the Relevant Distributions and Matrix Algebra”, 978-1138578333, Chapman and Hall/CRC, 1 edition, March 2018.↗

Similar topics

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

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Chi square distribution – demystified

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A random variable is always associated with a probability distribution. When the random variable undergoes mathematical transformation the underlying probability distribution no longer remains the same. Consider a random variable whose probability distribution function (PDF) is a standard normal distribution ( and ). Now, if the random variable is squared (a mathematical transformation), then the PDF of is no longer a standard normal distribution. The new transformed distribution is called Chi square Distribution with degree of freedom. The PDF of and are plotted in Figure 1.

Figure 1: Transformation of Normal Distribution to Chi Square Distribution

The mean of the random variable is and for the transformed variable Z2, the mean is given by . Similarly, the variance of the random variable is , whereas the variance of the transformed random variable is . In addition to the mean and variance, the shape of the distribution is also changed. The distribution of the transformed variable is no longer symmetric. In fact, the distribution is skewed to one side. Also the random variable can take only positive values whereas the random variable can take negative values too (note the x-axis in the plots above).

Since the new transformation is based on only one parameter (), the degree of freedom for this transformation is . Therefore, the transformed random variable follows – “Chi-square distribution with degree of freedom”.
Suppose, if are independent random variables that follows standard normal distribution( and ), then the transformation,

is a Chi square distribution with k degrees of freedom. The following figure illustrates how the definition of the Chi square distribution as a transformation of normal distribution for degree of freedom and degrees of freedom. In the same manner, the transformation can be extended to degrees of freedom.

Figure 2: Illustration of Chi-square Distribution with 2 degrees of freedom

The above equation is derived from random variables that follow standard normal distribution. For a standard normal distribution, the mean . Therefore, the transformation is called central Chi-square distribution. If, the underlying random variables follow normal distribution with non-zero mean, then the transformation is called non-central Chi-square distribution [2] . In channel modeling, the central Chi-squared distribution is related to Rayleigh Fading scenario and the non-central Chi-square distribution is related to Rician Fading scenario.

Mathematically, the PDF of the central Chi-squared distribution with degrees of freedom is given by

The mean and variance of the central Chi-squared distributed random variable is given by

Relation to Rayleigh distribution

The connection between Chi square distribution and the Rayleigh distribution can be established as follows

  1. If a random variable has standard Rayleigh distribution, then the transformation follows chi-square distribution with degrees of freedom.
  2. If a random variable has the chi-square distribution with degrees of freedom, then the transformation has standard Rayleigh distribution.

Applications:

Chi-square distribution is used in hypothesis testing (to compare the observed data with expected data that follows a specific hypothesis) and in estimating variances of a parameter.

Matlab Simulation:

Check this book for full Matlab code.
Wireless Communication Systems using Matlab – by Mathuranathan Viswanathan

Figure 3: Simulated output – central Chi square Distribution with k degrees of freedom

Python Code

Python numpy package has a chisquare() generator, which can be used in a straightforward manner to obtain the Chi square distributed sequences.

#---------Chi square distribution gaussianwaves.com-----
import numpy as np
import matplotlib.pyplot as plt
#%matplotlib inline
plt.style.use('ggplot')

ks=np.arange(start=1,stop=6,step=1) #degrees of freedoms to simulate
nSamp=1000000 #number of samples to generate

fig, ax = plt.subplots(ncols=1, nrows=1, constrained_layout=True)

for i,k in enumerate(ks):
    #Generate central Chi-square distributed random numbers
    X = np.random.chisquare(df=k, size = nSamp)
    ax.hist(X,bins=500,density=True,label=r'$k$={}'.format(k), \
    histtype='step',alpha=0.75, linewidth=3)

ax.set_xlim(left=0,right=8);ax.set_ylim(bottom=0,top=0.5);ax.legend();
ax.set_title('PDFs of Chi square distribution');
ax.set_xlabel(r'$\chi_k^2$');ax.set_ylabel(r'$f_{\chi_k^2}(x)$');
plt.show()

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For further reading

[1] Ernie Croot, “Notes on Chi-squared distribution”, Georgia institute of technology, School of mathematics, Oct 2005.↗

Similar topics

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
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Digital Modulations using Python
(PDF ebook)
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Uniform random variable

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Uniform random variables are used to model scenarios where the expected outcomes are equi-probable. For example, in a communication system design, the set of all possible source symbols are considered equally probable and therefore modeled as a uniform random variable.

The uniform distribution is the underlying distribution for an uniform random variable. A continuous uniform random variable, denoted as , take continuous values within a given interval , with equal probability. Therefore, the PDF of such a random variable is a constant over the given interval is.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

$$  f_X(x) = \begin{cases}\frac{1}{b-a} & \text{when } a < x < b\\0 & \text{otherwise} \end{cases} $$

In Matlab, rand function generates continuous uniform random numbers in the interval . The rand function picks a random number in the interval in which the probability of occurrence of all the numbers in the interval are equally likely. The command rand(n,m) will generate a matrix of size . To generate a random number in the interval one can use the following expression.

a + (b-a)*rand(n,m); %Here nxm is the size of the output matrix

To test whether the numbers generated by the continuous uniform distribution are uniform in the interval , one has to generate very large number of values using the rand function and then plot the histogram. The Figure 1 shows that the simulated PDF and theoretical PDF are in agreement with each other.

a=2;b=10; %open interval (2,10)
X=a+(b-a)*rand(1,1000000);%simulate uniform RV
[p,edges]=histcounts(X,'Normalization','pdf');%estimated PDF
outcomes = 0.5*(edges(1:end-1) + edges(2:end));%possible outcomes
g=1/(b-a)*ones(1,length(outcomes)); %Theoretical PDF
bar(outcomes,p);hold on;plot(outcomes,g,'r-');
title('Probability Density Function');legend('simulated','theory');
xlabel('Possible outcomes');ylabel('Probability of outcomes');
Figure 1: Continuous uniform random variable : histogram and theoretical PDF

On the other hand, a discrete random variable generates discrete values that are equally probable. The underlying discrete uniform distribution is denoted as , where , is a finite set of discrete elements that are equally probable as described by the probability mass function (PMF)

$$ f_X(x)= \begin{cases}\frac{1}{n} & \text{where } x \in {s_1,s_2,…,s_n } \\ 0 & otherwise \end{cases} $$

There exist several methods to generate discrete uniform random numbers and two of them are discussed here. The straightforward method is to use randi function in Matlab that can generate discrete uniform numbers in the integer set . The second method is to use rand function and ceil the result to discrete values. For example, the command to generate uniformly distributed discrete numbers from the set is

X=ceil(n*rand(1,100));

The uniformity test for discrete uniform random numbers can be performed and it is very similar to the code shown for the continuous uniform random variable case. The only difference here is the normalization term. The histogram values should not be normalized by the total area under the histogram curve as in the case of continuous random variables. Rather, the histogram should be normalized by the total number of occurrences in all the bins. We cannot normalized based on the area under the curve, since the bin values are not dense enough (bins are far from each other) for proper calculation of total area. The code snippet is given next. The resulting plot (Figure 2) shows a good match between the simulated and theoretical PMFs.

X=randi(6,100000,1); %Simulate throws of dice,S={1,2,3,4,5,6}
[pmf,edges]=histcounts(X,'Normalization','pdf');%estimated PMF
outcomes = 0.5*(edges(1:end-1) + edges(2:end));%S={1,2,3,4,5,6}
g=1/6*ones(1,6); %Theoretical PMF
bar(outcomes,pmf);hold on;stem(outcomes,g,'r-');
title('Probability Mass Function');legend('simulated','theory');
xlabel('Possible outcomes');ylabel('Probability of outcomes');
Figure 2: Discrete uniform random variable : histogram and theoretical PMFs
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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
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Checkout Added to cart
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