Random Variables Archives

Statistical measures for stochastic signals

Key focus: Discuss statistical measures for stochastic signals : mean, variance, skewness, kurtosis, histogram, scatterplot, cross-correlation and auto-correlation.

Deterministic and stochastic signals

A deterministic signal is exactly predictable for the given time span of interest. It could be expressed using analytic form (example: x(t) = sin (2 π f_c t) ).

Many of the signals that are encountered in real world signal processing applications cannot be expressed or predicted using analytic equations, because they have an element of uncertainty associated with them. As a consequence, such signals are characterized using statistical concepts. Therefore, these signals are outside the realm of deterministic signals and are classified as stochastic signals.

For example, we look at an electrical circuit and monitor the voltage across a resistor for a period of time. Under an applied electric field, atomic particles inside resister tend to randomly move and it manifests as heat. This random thermal motion causes random fluctuation in the voltage measured across the resistor. Therefore, the measured voltage can be characterized by a probability distribution and can be analyzed using statistical methods, but it cannot be predicted with precision. This is an example of signal that is stochastic function of time. Such functions usually evolve according to certain probabilistic laws and are assumed to be generated by an underlying stochastic process (thermal motion in the example above).

*Figure 1: Examples for deterministic and stochastic signals*

*Figure 1: Examples for deterministic and stochastic signals*

Given the amplitude of the stochastic voltage signal at time , now we know, we cannot predict the value at t’. However, if we observed the signal for a sufficient amount of time, we can empirically determine its probability distribution based on which we should be able to answer questions like

Given the amplitude of the voltage at time t, what is the average (expected or mean) of the voltage at time t’?
How much can we expect the voltage at time t’, to fluctuate from the mean ? In other words, we are interested in the variance of the voltage at time t’.
What is the probability that the voltage at t’ exceeds or falls below a certain threshold value ?
How the voltages at times t and t’ are related ? In other words, we are interested in correlation.

Summary of descriptive statistical measures

Different statistical measures are available to gather, review, analyze and draw conclusions from stochastic signals. Some of the statistical measures are summarized below:

Quantitative measures of shape:

In many statistical analysis, the fundamental task is to get a general idea about the shape of a distribution and it is done by using moments.

Measure of central tendency – mean – the first moment:

Measures of central tendency attempt to identify the central position in the distribution of samples that make up the stochastic signal. Mean, mode and median are different measures of central tendency. In signal processing, we are mostly interested in computing mean which is the average of values of the given samples. Given a discrete random variable X with probability mass function p_X(x) the mean is denoted by

\[\mu = E \left[ X\right] = \sum_{x: p_X(x) > 0} x p_X(x) \]

Measure of dispersion – variance – the second moment:

Measures of dispersion describe how the values in the signal samples are spread out around a central location. Variance and standard deviation are some of the measures of dispersion. If the values are widely dispersed, the central location is said to be less representative of the values as a whole. If the values are tightly dispersed, the central location is considered more reliable.

For a discrete random variable X with probability mass function p_X(x) , the variance (σ_X²) is given by the second central moment. The term central moment implies that this is measured relative to mean. For an electrical signal, the second moment is proportional to the average power.

\[\sigma_X^2 = E \left[ \left(X – \mu \right)^2\right] = \sum_{x: p_X(x) > 0} \left(x – \mu \right)^2 p_X(x) \]

The square root of variance is standard deviation

\[\sigma_X = \sqrt{E \left[ \left(X – \mu \right)^2\right]}\]

Figure 2, demonstrates the use of histogram for getting a visual perception of shape of the distribution of samples from two different stochastic signals. Though the signals look similar in nature in time domain, their histogram reveals different picture altogether. The central location (mean) of the first signal is around zero (no DC shift in the time domain view) and for the second signal the mean is at 0.75. The average power of first signal varies widely (histogram is widely spread out) compared to that of the second signal.

*Figure 2: Histogram is a visual method that provides a general idea about the shape of a distribution.*

Higher order moments – skewness and kurtosis:

Further characterization of the shape of a distribution includes higher order moments : skewness and kurtosis. They identify anomalies and outliers in many signal processing applications.

Skewness provides a measure to quantify the presence of asymmetry in the shape of the distribution. Actually, skewness measures the relative size of the tails in a distribution. Presence of asymmetry manifests as non-zero value for the standardized third central moment. The term “standardized” stands for the normalization of the third central moment by σ³.

\[Skewness = \frac{E \left[ \left( X – \mu \right)^3 \right]}{\sigma^3}\]

*Figure 3: A random sequence showing positive skewness (asymmetry in shape) in its distribution*

Kurtosis measures the amount of probability in the two tails of a distribution, relative to a normal distribution of same variance. Kurtosis is 3 for normal distribution (perfect bell shaped curve). If the kurtosis of a distribution is greater than 3, it implies that the tails are heavier compared to that of normal distribution. A value less than 3 for kurtosis implies lighter tails compared to that of the normal distribution. Essentially, kurtosis is a measure of outliers. Kurtosis is calculated as the standardized fourth central moment.

\[Kurtosis = \frac{E \left[ \left( X – \mu \right)^4 \right]}{\sigma^4}\]

*Figure 4: Histogram showing kurtosis of a random sequence vs. kurtosis of normal distribution*

Measures of association

Statistics, such as measures of central tendency, dispersion and higher order moments, describe about a single distribution are called univariate statistics. If we are interested in the relationship between two or more variables, we have to move to at least the realm of bivariate statistics.

Measures of association, summarize the size of association between two variables. Most measures of associates are scaled to a range of values. For example, a measure of association can be construed to have a range 0 to 1, where the value 0 implies no relationship and a value of 1 implies perfect relationship between the two variables under study. In another case, the measure can range from -1 to 1, which can help us determine if the two variables have negative or positive relationship between each other.

Scatter plot

Scatter plot is a great way to visually assess the nature of relationship between two variables. Following figure contains the scatter plots between different stochastic signals, exhibiting different strengths of relationships between the signals.

Correlation

Correlation functions are commonly used in signal processing, for measuring the similarity of two signals. They are especially used in signal detection and pattern recognition.

Cross-correlation

Cross-correlation is commonly used for measuring the similarity between two different signals. In signal processing, cross-correlation measures the similarity of two waveforms as a function of time lags applied to one of the waveform.

For discrete-time waveforms – x[n] and y[n], cross correlation is defined as

\[Corr_{xy}[l] = \sum_{m=-\infty}^{\infty} x[n]^{\ast} y[n+l] = \sum_{m=-\infty}^{\infty} x[n-l]^{\ast} y[n]\]

where, * denotes complex conjugate operation and l is the discrete time lags applied to one of the waveforms. That is, the cross-correlation can be calculated as the dot product of a sequence with each shifted version of another sequence.

Auto-correlation

Auto-correlation is the cross-correlation of a waveform/sequence with itself.

For discrete-time waveform – x[n], auto-correlation is defined as

\[Corr_{xx}[l] = \sum_{m=-\infty}^{\infty} x[n]^{\ast} x[n+l] = \sum_{m=-\infty}^{\infty} x[n-l]^{\ast} x[n] \]

where, * denotes complex conjugate operation and l is the discrete time lags applied to the copy of the same waveform.

Correlation properties are useful for identifying/distinguishing a known bit sequence from a set of other possible known sequences. For example, in GPS technology, each satellite is assigned a unique 10-bit Gold code sequence (2¹⁰ = 1023 possible combinations). Cross-correlation between different Gold code sequence is very low, since the Gold codes are orthogonal to each other. However, the auto-correlation of a Gold code is maximum at zero lag. The satellites are identified using these correlation properties of Gold codes. (I have described the hardware implementation of Gold codes, it can be accessed here).

Following plots illustrate the application of auto-correlation for audio analysis. The auto-correlation of an audio signal will have a peak at zero lag (i.e, where there is no time shifting when computing the correlation) as shown in Figure 6. Figure 7 contains the same audio file that is synthetically processed to produce reverberation characteristics. By looking at the time series plot in Figure 7, we cannot infer anything. However, the auto-correlation plot reveals the reverberation characteristics embedded in the audio.

*Figure 6: Normalized auto-correlation of an original sound*

*Figure 7: Normalized auto-correlation of sound with simulated reverberation characteristics (synthetically processed through an IIR comb filter)*

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Wireless Communication Systems in Matlab Second Edition(PDF) (172 votes, average: 3.66 out of 5) Checkout Added to cart	Digital Modulations using Python (PDF ebook) (127 votes, average: 3.58 out of 5) Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) (134 votes, average: 3.63 out of 5) Checkout Added to cart
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For further reading

[1] Steven M. Kay, “Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory”, ISBN: 978-0133457117, Prentice Hall, Edition 1, 1993.↗

Matplotlib histogram and estimated PDF in Python

Key focus: Shown with examples: let’s estimate and plot the probability density function of a random variable using Python’s Matplotlib histogram function.

Generation of random variables with required probability distribution characteristic is of paramount importance in simulating a communication system. Let’s see how we can generate a simple random variable, estimate and plot the probability density function (PDF) from the generated data and then match it with the intended theoretical PDF. Normal random variable is considered here for illustration.

Step 1: Generate random samples

A survey of commonly used fundamental methods to generate a given random variable is given in [1]. For this demonstration, we will consider the normal random variable with the following parameters : μ – mean and σ – standard deviation. First generate a vector of randomly distributed random numbers of sufficient length (say 100000) with some valid values for μ and σ. There are more than one way to generate this. Two of them are given below.

● Method 1: Using the in-built numpy.random.normal() function (requires numpy package to be installed)

import numpy as np

mu=10;sigma=2.5 #mean=10,deviation=2.5
L=100000 #length of the random vector

#Random samples generated using numpy.random.normal()
samples_normal = np.random.normal(loc=mu,scale=sigma,size=(L,1)) #generate normally distributted samples

● Method 2: Box-Muller transformation [2] method produces a pair of normally distributed random numbers (Z₁, Z₂) by transforming a pair of uniformly distributed independent random samples (U₁,U₂). The algorithm for transformation is given by

#Samples generated using Box-Muller transformation

U1 = np.random.uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
U2 = np.random.uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)

a = np.sqrt(-2*np.log(U1))
b = 2*np.pi*U2

Z = a*np.cos(b) #Standard Normal distributed numbers
samples_box_muller= Z*sigma+mu #Normal distribution with mean and sigma

Step 2: Plot the estimated histogram

Typically, if we have a vector of random numbers that is drawn from a distribution, we can estimate the PDF using the histogram tool. Matplotlib’s hist function can be used to compute and plot histograms. If the density argument is set to ‘True’, the hist function computes the normalized histogram such that the area under the histogram will sum to 1. Estimate and plot the normalized histogram using the hist function.

#For plotting
import matplotlib.pyplot as plt
%matplotlib inline
plt.style.use('ggplot')

fig, ax0 = plt.subplots(ncols=1, nrows=1) #creating plot axes
(values, bins, _) = ax0.hist(samples_normal,bins=100,density=True,label="Histogram of samples") #Compute and plot histogram, return the computed values and bins

Step 3: Theoretical PDF:

And for verification, overlay the theoretical PDF for the intended distribution. The theoretical PDF of normally distributed random samples is given by

Theoretical PDF for normal distribution is readily obtained from stats.norm.pdf() function in the SciPy package.

from scipy import stats
bin_centers = 0.5*(bins[1:] + bins[:-1])
pdf = stats.norm.pdf(x = bin_centers, loc=mu, scale=sigma) #Compute probability density function
ax0.plot(bin_centers, pdf, label="PDF",color='black') #Plot PDF
ax0.legend()#Legend entries
ax0.set_title('PDF of samples from numpy.random.normal()');

*Figure 1: Estimated PDF (histogram) and the theoretical PDF for samples generated using numpy.random.normal() function*

The histogram and theoretical PDF of random samples generated using Box-Muller transformation, can be plotted in a similar manner.

#Samples generated using Box-Muller transformation
from numpy.random import uniform
U1 = uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
U2 = uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
a = np.sqrt(-2*np.log(U1))
b = 2*np.pi*U2
Z = a*np.cos(b) #Standard Normal distribution
samples_box_muller = Z*sigma+mu #Normal distribution with mean and sigma

#Plotting
fig, ax1 = plt.subplots(ncols=1, nrows=1) #creating plot axes
(values, bins, _) = ax1.hist(samples_box_muller,bins=100,density=True,label="Histogram of samples") #Plot histogram
bin_centers = 0.5*(bins[1:] + bins[:-1])
pdf = stats.norm.pdf(x = bin_centers, loc=mu, scale=sigma) #Compute probability density function
ax1.plot(bin_centers, pdf, label="PDF",color='black') #Plot PDF
ax1.legend()#Legend entries
ax1.set_title('Box-Muller transformation');

*Figure 2: Estimated PDF (histogram) and theoretical PDF for samples generated using Box-Muller transformation for normal distribution*

References:

[1] John Mount, ‘Six Fundamental Methods to Generate a Random Variable’, January 20, 2012
[2] Thomas, D. B., Luk. W., Leong, P. H. W., and Villasenor, J. D. 2007. Gaussian random number generators. ACM Comput. Surv. 39, 4, Article 11 (October 2007), 38 pages DOI = 10.1145/1287620.1287622 http://doi.acm.org/10.1145/1287620.1287622

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Articles in this series
[1] Fibonacci series in python
[2] Central Limit Theorem – a demonstration
[3] Moving Average Filter in Python and Matlab
[4] How to plot FFT in Python – FFT of basic signals : Sine and Cosine waves
[5] How to plot audio files as time-series using Scipy python
[6] How to design a simple FIR filter to reject unwanted frequencies
[7] Analytic signal, Hilbert Transform and FFT
[8] Non-central Chi-squared Distribution
[9] Simulation of M-PSK modulation techniques in AWGN channel (in Matlab and Python)
[10] QPSK modulation and Demodulation (with Matlab and Python implementation)

Plot histogram and estimated PDF in Matlab

Key focus: With examples, let’s estimate and plot the probability density function of a random variable using Matlab histogram function.

Note: If you are inclined towards programming in Python, visit this article

Step 1: Create the random variable

A survey of commonly used fundamental methods to generate a given random variable is given in [1]. For this demonstration, we will consider the normal random variable with the following parameters : – mean and – standard deviation. First generate a vector of randomly distributed random numbers of sufficient length (say 100000) with some valid values for and . There are more than one way to generate this. Some of them are given below.

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.

● Method 1: Using the in-built random function (requires statistics toolbox)

mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
R = random('Normal',mu,sigma,L,1);%method 1

● Method 2: Using randn function that generates normally distributed random numbers having and = 1

mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
R = randn(L,1)*sigma + mu; %method 2

● Method 3: Box-Muller transformation [2] method using rand function that generates uniformly distributed random numbers

 mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
U1 = rand(L,1); %uniformly distributed random numbers U(0,1)
U2 = rand(L,1); %uniformly distributed random numbers U(0,1)
Z = sqrt(-2log(U1)).cos(2piU2);%Standard Normal distribution
R = Z*sigma+mu;%Normal distribution with mean and sigma

Step 2: Plot the estimated histogram

Typically, if we have a vector of random numbers that is drawn from a distribution, we can estimate the PDF using the histogram tool. Matlab supports two in-built functions to compute and plot histograms:

● hist – introduced before R2006a
● histogram – introduced in R2014b

Which one to use ? Matlab’s help page points that the hist function is not recommended for several reasons and the issue of inconsistency is one among them. The histogram function is the recommended function to use.

Estimate and plot the normalized histogram using the recommended ‘histogram’ function. And for verification, overlay the theoretical PDF for the intended distribution. When using the histogram function to plot the estimated PDF from the generated random data, use ‘pdf’ option for ‘Normalization’ option. Do not use the ‘probability’ option for ‘Normalization’ option, as it will not match the theoretical PDF curve.

histogram(R,'Normalization','pdf'); %plot estimated pdf from the generated data

X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory = pdf('Normal',X,mu,sigma); %theoretical normal probability density
hold on; plot(X,fx_theory,'r'); %plot computed theoretical PDF
title('Probability Density Function'); xlabel('values - x'); ylabel('pdf - f(x)'); axis tight;
legend('simulated','theory');

*Estimated PDF (using histogram function) and the theoretical PDF*

However, if you do not have Matlab version that was released before R2014b, use the ‘hist’ function and get the histogram frequency counts () and the bin-centers (). Using these data, normalize the frequency counts using the overall area under the histogram. Plot this normalized histogram and overlay the theoretical PDF for the chosen parameters.

%For those who don't have access to 'histogram' function
%get un-normalized values from hist function with same number of bins as histogram function
numBins=50; %choose appropriately
[f,x]=hist(R,numBins); %use hist function and get unnormalized values
figure; plot(x,f/trapz(x,f),'b-*');%plot normalized histogram from the generated data

X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory =   pdf('Normal',X,mu,sigma); %theoretical normal probability density
hold on; plot(X,fx_theory,'r'); %plot computed theoretical PDF
title('Probability Density Function'); xlabel('values - x'); ylabel('pdf - f(x)'); axis tight;
legend('simulated','theory');

*Estimated PDF (using hist function) and the theoretical PDF*

Step 3: Theoretical PDF:

The given code snippets above, already include the command to plot the theoretical PDF by using the ‘pdf’ function in Matlab. It you do not have access to this function, you could use the following equation for computing the theoretical PDF

The code snippet for that purpose is given next.

X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory = 1/sqrt(2*pi*sigma^2)*exp(-0.5*(X-mu).^2./sigma^2);
plot(X,fx_theory,'k'); %plot computed theoretical PDF

Note: The functions – ‘random’ and ‘pdf’ , requires statistics toolbox.

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References:

[1] John Mount, ‘Six Fundamental Methods to Generate a Random Variable’, January 20, 2012.↗
[2] Thomas, D. B., Luk. W., Leong, P. H. W., and Villasenor, J. D. 2007. Gaussian random number generators. ACM Comput. Surv. 39, 4, Article 11 (October 2007), 38 pages DOI = 10.1145/1287620.1287622 http://doi.acm.org/10.1145/1287620.1287622.↗

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

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Generate multiple sequences of correlated random variables

In the previous post, a method for generating two sequences of correlated random variables was discussed. Generation of multiple sequences of correlated random variables, given a correlation matrix is discussed here.

Correlation Matrix

Correlation matrix defines correlation among N variables. It is a symmetric matrix with the element equal to the correlation coefficient between the and the variable. The diagonal elements (correlations of variables with themselves) are always equal to 1.

Sample problem:

Let’s say we would like to generate three sets of random sequences X,Y,Z with the following correlation relationships.

Correlation co-efficient between X and Y is 0.5
Correlation co-efficient between X and Z is 0.3
Obviously the variable X correlates with itself 100% – i.e, correlation-coefficient is 1

Putting all these relationships in a compact matrix form, gives the correlation matrix. We take arbitrary correlation value (0.3) for the relationship between Y and Z.

Now, the task is to generate three sets of random numbers X,Y and Z that follows the relationship above. The problem can be addressed in many ways. Two most common methods finding the solution are

Cholesky Decomposition method
Spectral Decomposition ( also called Eigen Vector Decomposition) method

The Cholesky Decomposition method is discussed here.

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.

Generating Correlated random number using Cholesky Decomposition:

Cholesky decomposition is the matrix equivalent of taking square root operation on a given matrix. As with any scalar values, positive square root is only possible if the given number is a positive (Imaginary roots do exist otherwise). Similarly, if a matrix need to be decomposed into square-root equivalent, the matrix need to be positive definite.

The method discussed here, seeks to decompose the given correlation matrix using Cholesky decomposition.

where U and L are upper and lower triangular matrices. We will consider Upper triangular matrix here. Equivalently, lower triangular matrix can also be used, in which case the order of output needs to be reversed.

For this decomposition to work, the correlation matrix should be positive definite. The correlated random sequences (where X,Y,Z are column vectors) that follow the above relationship can be generated by multiplying the uncorrelated random numbers R with U .

Steps to follow:

Generate three sequences of uncorrelated random numbers R – each drawn from a normal distribution. For this case, the R matrix will be of size where k is the number of samples we wish to generate and we allocate the k samples in three columns, where the columns indicate the place holder for each variable X, Y and Z. Multiply this matrix with the Cholesky decomposed upper triangular version of the correlation matrix.

Python code

import numpy as np
from scipy.linalg import cholesky
from scipy.stats import pearsonr #to calculate correlation coefficient

#for plotting and visualization
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
plt.style.use('fivethirtyeight')
import seaborn as sns

C = np.array([[1, -0.5, 0.3], 
              [-0.5, 1, 0.2],
              [0.3, 0.2, 1]]) #Construct correlation matrix
U = cholesky(C) #Cholesky decomposition
R = np.random.randn(10000,3) #Three uncorrelated sequences
Rc = R @ U #Array of correlated random sequences

#compute and display correlation coeff from generated sequences
def pearsonCorr(x, y, **kws): 
    (r, _) = pearsonr(x, y) #returns Pearson’s correlation coefficient, 2-tailed p-value)
    ax = plt.gca()
    ax.annotate("r = {:.2f} ".format(r),xy=(.7, .9), xycoords=ax.transAxes)
    
#Visualization
df = pd.DataFrame(data=Rc, columns=['X','Y','Z'])
graph = sns.pairplot(df)
graph.map(pearsonCorr)

Figure 1: Pairplot of correlated random variables generated using Cholesky decomposition (Python)

Matlab code

x=[  1  0.5 0.3; 0.5  1  0.3; 0.3 0.3  1 ;]; %Correlation matrix
U=chol(x); %Cholesky decomposition 

R=randn(10000,3); %Random data in three columns each for X,Y and Z
Rc=R*U; %Correlated matrix Rc=[X Y Z]

%Verify Correlation-Coeffs of generated vectors
coeffMatrixXX=corrcoef(Rc(:,1),Rc(:,1));
coeffMatrixXY=corrcoef(Rc(:,1),Rc(:,2));
coeffMatrixXZ=corrcoef(Rc(:,1),Rc(:,3));

%Extract the required correlation coefficients
coeffXX=coeffMatrixXX(1,2) %Correlation Coeff for XX;
coeffXY=coeffMatrixXY(1,2) %Correlation Coeff for XY;
coeffXZ=coeffMatrixXZ(1,2) %Correlation Coeff for XZ;

%Scatterplots
subplot(3,1,1)
plot(Rc(:,1),Rc(:,1),'b.')
title(['Scatterd Plot - X and X calculated \rho=' num2str(coeffXX)])
xlabel('X')
ylabel('X')

subplot(3,1,2)
plot(Rc(:,1),Rc(:,2),'r.')
title(['Scatterd Plot - X and Y calculated \rho=' num2str(coeffXY)])
xlabel('X')
ylabel('Y')

subplot(3,1,3)
plot(Rc(:,1),Rc(:,3),'m.')
title(['Scatterd Plot - X and Z calculated \rho=' num2str(coeffXZ)])
xlabel('X')
ylabel('Z')

Scattered plots to verify the simulated data

Figure 2: Scatter plots showing the correlation trend between the three random variables (Matlab)

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Generate two correlated random sequences

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This article discusses the method of generating two correlated random sequences using Matlab. If you are looking for the method on generating multiple sequences of correlated random numbers, I urge you to go here.

Generating two vectors of correlated random numbers, given the correlation coefficient , is implemented in two steps. The first step is to generate two uncorrelated random sequences from an underlying distribution. Normally distributed random sequences are considered here.

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.

Step 1: Generate two uncorrelated Gaussian distributed random sequences

x1=randn(1,100); %Normal random numbers sequence 1
x2=randn(1,100); %Normal random numbers sequence 2
subplot(1,2,1); plot(x1,x2,'r*');
title('Uncorrelated RVs X_1 and X_2');
xlabel('X_1'); ylabel('X_2');

Step 2: Generate correlated random sequence z

In the second step, the required correlated sequence is generated as

rho=0.9;
z=rho*x1+sqrt(1-rhoˆ2)*x2;%transformation
subplot(1,2,2); plot(x1,z,'r*');
title(['Correlated RVs X_1 and Z , \rho=',num2str(rho)]);
xlabel('X_1'); ylabel('Z');

The resulting sequence Z will have correlation with respect to

Results plotted below.

*Figure : Scatter plots – Correlated random variables and on right*

Continue reading this article on the method to generate multiple vectors of correlated random numbers.

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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()

*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.↗

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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

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

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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.

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● 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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Random Variables, CDF and PDF

Random Variable:

In a “coin-flipping” experiment, the outcome is not known prior to the experiment, that is we cannot predict it with certainty (non-deterministic/stochastic). But we know the all possible outcomes – Head or Tail. Assign real numbers to the all possible events (this is called “sample space”), say “0” to “Head” and “1” to “Tail”, and associate a variable “X” that could take these two values. This variable “X” is called a random variable, since it can randomly take any value ‘0’ or ‘1’ before performing the actual experiment.

Obviously, we do not want to wait till the coin-flipping experiment is done. Because the outcome will lose its significance, we want to associate some probability to each of the possible event. In the coin-flipping experiment, all outcomes are equally probable (given that the coin is fair and unbiased). This means that we can say that the probability of getting Head ( our random variable X = 0 ) as well that of getting Tail ( X =1 ) is 0.5 (i.e. 50-50 chance for getting Head/Tail).

This can be written as,

Cumulative Distribution Function:

Mathematically, a complete description of a random variable is given be “Cumulative Distribution Function”- F_X(x). Here the bold faced “X” is a random variable and “x” is a dummy variable which is a place holder for all possible outcomes ( “0” and “1” in the above mentioned coin flipping experiment). The Cumulative Distribution Function is defined as,

If we plot the CDF for our coin-flipping experiment, it would look like the one shown in the figure on your right.
The example provided above is of discrete nature, as the values taken by the random variable are discrete (either “0” or “1”) and therefore the random variable is called Discrete Random Variable.

If the values taken by the random variables are of continuous nature (Example: Measurement of temperature), then the random variable is called Continuous Random Variable and the corresponding cumulative distribution function will be smoother without discontinuities.

Probability Distribution function :

Consider an experiment in which the probability of events are as follows. The probabilities of getting the numbers 1,2,3,4 individually are respectively. It will be more convenient for us if we have an equation for this experiment which will give these values based on the events. For example, the equation for this experiment can be given by where . This equation ( equivalently a function) is called probability distribution function.

Probability Density function (PDF) and Probability Mass Function(PMF):

Its more common deal with Probability Density Function (PDF)/Probability Mass Function (PMF) than CDF.

The PDF (defined for Continuous Random Variables) is given by taking the first derivate of CDF.

For discrete random variable that takes on discrete values, is it common to defined Probability Mass Function.

The previous example was simple. The problem becomes slightly complex if we are asked to find the probability of getting a value less than or equal to 3. Now the straight forward approach will be to add the probabilities of getting the values which comes out to be . This can be easily modeled as a probability density function which will be the integral of probability distribution function with limits 1 to 3.

Based on the probability density function or how the PDF graph looks, PDF fall into different categories like binomial distribution, Uniform distribution, Gaussian distribution, Chi-square distribution, Rayleigh distribution, Rician distribution etc. Out of these distributions, you will encounter Gaussian distribution or Gaussian Random variable in digital communication very often.

Mean:

The mean of a random variable is defined as the weighted average of all possible values the random variable can take. Probability of each outcome is used to weight each value when calculating the mean. Mean is also called expectation (E[X])

For continuos random variable X and probability density function f_X(x)

For discrete random variable X, the mean is calculated as weighted average of all possible values (x_i) weighted with individual probability (p_i)

Variance :

Variance measures the spread of a distribution. For a continuous random variable X, the variance is defined as

For discrete case, the variance is defined as

Standard Deviation () is defined as the square root of variance

Properties of Mean and Variance:

For a constant – “c” following properties will hold true for mean

For a constant – “c” following properties will hold true for variance

PDF and CDF define a random variable completely. For example: If two random variables X and Y have the same PDF, then they will have the same CDF and therefore their mean and variance will be same.
On the otherhand, mean and variance describes a random variable only partially. If two random variables X and Y have the same mean and variance, they may or may not have the same PDF or CDF.

Gaussian Distribution :

Gaussian PDF looks like a bell. It is used most widely in communication engineering. For example , all channels are assumed to be Additive White Gaussian Noise channel. What is the reason behind it ? Gaussian noise gives the smallest channel capacity with fixed noise power. This means that it results in the worst channel impairment. So the coding designs done under this most adverse environment will give superior and satisfactory performance in real environments. For more information on “Gaussianity” refer [1]

The PDF of the Gaussian Distribution (also called as Normal Distribution) is completely characterized by its mean () and variance(),

Since PDF is defined as the first derivative of CDF, a reverse engineering tell us that CDF can be obtained by taking an integral of PDF.
Thus to get the CDF of the above given function,

Equations for PDF and CDF for certain distributions are consolidated below

Probability Distribution	Probability Density Function(PDF)	Cumulative Distribution Function (CDF)
Gaussian/Normal Distribution –

Reference :

[1] S.Pasupathy, “Glories of Gaussianity”, IEEE Communications magazine, Aug 1989 – 1, pp 38.

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Deterministic and stochastic signals

Summary of descriptive statistical measures

Quantitative measures of shape:

Measure of central tendency – mean – the first moment:

Measure of dispersion – variance – the second moment:

Higher order moments – skewness and kurtosis:

Measures of association

Scatter plot

Correlation

Cross-correlation

Auto-correlation

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Step 1: Generate random samples

Step 2: Plot the estimated histogram

Step 3: Theoretical PDF:

References:

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Step 1: Create the random variable

Step 2: Plot the estimated histogram

Step 3: Theoretical PDF:

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Correlation Matrix

Sample problem:

Generating Correlated random number using Cholesky Decomposition:

Steps to follow:

Python code

Matlab code

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Step 1: Generate two uncorrelated Gaussian distributed random sequences <img decoding="async" src="https://s0.wp.com/latex.php?latex=X_1%2C+X_2&amp;bg=ffffff&amp;fg=000&amp;s=0&amp;c=20201002" alt="X_1, X_2" width="46" height="16">

Step 2: Generate correlated random sequence z

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

Matlab Code:

Python Code:

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Relation to Rayleigh distribution

Applications:

Matlab Simulation:

Python Code

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Random Variable:

Cumulative Distribution Function:

Probability Distribution function :

Probability Density function (PDF) and Probability Mass Function(PMF):

Mean:

Variance :

Properties of Mean and Variance:

Gaussian Distribution :

Reference :

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Step 1: Generate two uncorrelated Gaussian distributed random sequences