Tag: Random Process

Bayes’ theorem

Key focus: Bayes’ theorem is a method for revising the prior probability for specific event, taking into account the evidence available about the event.

Introduction

In statistics, the process of drawing conclusions from data subject to random variations – is called “statistical inference”. Usually, in any random experiment, the observations are recorded and conclusions have to be drawn based on the recorded data set. Conclusions over the underlying random process are necessary to establish one or many of the following:

* Estimation of a parameter of interest (For example: the carrier frequency estimation in the receiver)
* Confidence and credibility of the estimate
* Rejecting a preconceived hypothesis
* Classification of data set into groups

Several schools of statistical inference have evolved over time. Bayesian inference is one of them.

Bayes’ theorem

Bayes’ theorem is central to scientific discovery and a core tool in machine learning/AI. It has numerous applications including but not limited to areas such as: mathematics, medicine, finance, marketing and engineering.

The Bayes’ theorem is used in Bayesian inference, usually dealing with a sequence of events, as new information becomes available about a subsequent event, that new information is used to update the probability of the initial event. In this context, we encounter two flavors of probabilities: prior probability and posterior probability.

Prior probability : This is the initial probability about an event before any information is available about the event. In other words, this is the initial belief about a particular hypothesis before any evidence is available about the hypothesis.

Posterior probability: This is the probability value that has been revised by using new information that is later obtained from a subsequent event. In other words, this is the updated belief about the hypothesis as new evident becomes available.

The formula for Bayes’ theorem is

Figure 1: Formula for Bayes’ theorem

A very simple thought experiment

You are asked to conduct a random experiment with a given coin. You are told that the coin is unbiased (probability of obtaining head or tail is equal and is exactly 50%). You believe (before conducting the experiment) that the coin is unbiased and that the chance of getting head or tail is equal to be 0.5.

Assume that you have not looked at both sides of the coin and simply you start to conduct the experiment. You start to toss the coin repeatedly and record the events (This is the observed new information/evidences). On the first toss you observe the coin lands on the ground with head faced up. On the second toss, again the head shows up. On subsequent tosses, the coin always shows up head. You have tossed 100 times and all these tosses you observe only head. Now what will you think about the coin? You will really start to think that both sides of the coin are engraved with “head” (no tail etched on the coin). Now, based on the new evidences, your belief about the “unbiasedness” of the coin is altered.

This is what Bayes’ theorem or Bayesian inference is all about. It is a general principle about learning from experience. It connects beliefs (called prior probabilities) and evidences (observed data). Based on the evidence, the degree of belief is refined. The degree of belief after conducting the experiment is called posterior probability.

Figure 2: Bayes’ theorem – the process

Real world example

Suppose, a person X falls sick and goes to the doctor for diagnosis. The doctor runs a series of tests and the test result came positive for a rare disease that affects 0.1% of the population. The accuracy of the test is 99%. That is, the test can correctly identify 99% of people that have the disease and will incorrectly report disease in only 1% of the people that do not have the disease. Now, how certain is that the person X actually have the disease ?

In this scenario, we can apply the extended form of Bayes’ theorem

Figure 3: Bayes’ theorem – extended form

Extended form of Bayes’ theorem is applied in special scenarios where P(H) is a binary variable, which implies it can take only two possible states. In the given problem above, the hypothesis can take only two states – H – “having the disease” and H̅ – “not having the disease”.

For the given problem, we can come up with the following numbers for the various quantities in the extended form of Bayes’ theorem.

P(H) = prior probability of having the disease before the availability of test results. This is often guess work, but luckily we have the probability that affects the population (0.1% = 0.001) to replace this.
P(E/H) = probability to test positive for the disease if person X has the disease (99% = 0.99)
P(H̅) = probability of NOT having the disease (1-0.001 = 0.999)
P(E/H̅) = probability of NOT having the disease and falsely identified positive by the test (1% = 0.01).
P(H/E) = probability of person X actually have the disease given the test result is positive.

Plugging-in these numbers in the extended form of Bayes’ theorem, we get the probability that X actually have the disease is just 9%.

Figure 4: Calculation using extended form of Bayes’ theorem

Person X doubts the result and goes for a second opinion to another doctor and gets tested from an independent laboratory. The second test result came back positive this time too. Now what is the probability that person X actually have the disease ?

P(H) = Replace this with the posterior probability from first test (we are refining the belief about the result of the first test) = 9.016% = 0.09016
P(E/H) = probability to test positive for the disease if person X has the disease (99% = 0.99)
P(H̅) = probability of NOT having the disease from first test (1-0.09016 = 0.90984)
P(E/H̅) = probability of NOT having the disease and falsely identified positive by the second test (1% = 0.01).
P(H/E) = probability of person X actually have the disease given the second test result is also positive.

Figure 5: Refining the belief about the first test using results from second test

Therefore, the updated probability based on two positive tests is 90.75%. This implies that there is a 90.75% chance that person X has the disease.

I hope the reader got a better understanding of what Bayes’ theorem is, various parameters in the equation for Bayes’ theorem and how to apply it.

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References

[1] Jeremy Orloff and Jonathan Bloom, “Conditional Probability, Independence and Bayes’ Theorem”, MIT OCW, Class 3, 18.05 Introduction to Probability and Statistics ↗.
[2] Veritasium, “The Bayesian Trap”, YouTube

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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.

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. Other types of random variables like uniform, Bernoulli, binomial, Chi-squared, Nakagami-m are illustrated in the next section.

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.

  1. Correlation co-efficient between X and Y is 0.5
  2. Correlation co-efficient between X and Z is 0.3
  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

  1. Cholesky Decomposition method
  2. 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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Further reading

[1] Richard Taylor, “Interpretation of correlation coefficient: A basic review”, Journal of diagnostic medical sonography, Jan/Feb 1990.↗

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 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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Further reading

[1] Richard Taylor, “Interpretation of correlation coefficient: A basic review”, Journal of diagnostic medical sonography, Jan/Feb 1990.↗

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
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Derive BPSK BER – optimum receiver in AWGN channel

Key focus: Derive BPSK BER (bit error rate) for optimum receiver in AWGN channel. Explained intuitively step by step.

BPSK modulation is the simplest of all the M-PSK techniques. An insight into the derivation of error rate performance of an optimum BPSK receiver is essential as it serves as a stepping stone to understand the derivation for other comparatively complex techniques like QPSK,8-PSK etc..

Understanding the concept of Q function and error function is a pre-requisite for this section of article.

The ideal constellation diagram of a BPSK transmission (Figure 1) contains two constellation points located equidistant from the origin. Each constellation point is located at a distance from the origin, where Es is the BPSK symbol energy. Since the number of bits in a BPSK symbol is always one, the notations – symbol energy (Es) and bit energy (Eb) can be used interchangeably (Es=Eb).

Assume that the BPSK symbols are transmitted through an AWGN channel characterized by variance = N0/2 Watts. When 0 is transmitted, the received symbol is represented by a Gaussian random variable ‘r‘ with mean=S0 = and variance =N0/2. When 1 is transmitted, the received symbol is represented by a Gaussian random variable – r with mean=S1= and variance =N0/2. Hence the conditional density function of the BPSK symbol (Figure 2) is given by,

Figure 1: BPSK – ideal constellation
Figure 2: Probability density function (PDF) for BPSK Symbols

 An optimum receiver for BPSK can be implemented using a correlation receiver or a matched filter receiver (Figure 3). Both these forms of implementations contain a decision making block that decides upon the bit/symbol that was transmitted based on the observed bits/symbols at its input.

Figure 3: Optimum Receiver for BPSK

When the BPSK symbols are transmitted over an AWGN channel, the symbols appears smeared/distorted in the constellation depending on the SNR condition of the channel. A matched filter or that was previously used to construct the BPSK symbols at the transmitter. This process of projection is illustrated in Figure 4. Since the assumed channel is of Gaussian nature, the continuous density function of the projected bits will follow a Gaussian distribution. This is illustrated in Figure 5.

Figure 4: Role of correlation/Matched Filter

After the signal points are projected on the basis function axis, a decision maker/comparator acts on those projected bits and decides on the fate of those bits based on the threshold set. For a BPSK receiver, if the a-prior probabilities of transmitted 0’s and 1’s are equal (P=0.5), then the decision boundary or threshold will pass through the origin. If the apriori probabilities are not equal, then the optimum threshold boundary will shift away from the origin.

Figure 5: Distribution of received symbols

Considering a binary symmetric channel, where the apriori probabilities of 0’s and 1’s are equal, the decision threshold can be conveniently set to T=0. The comparator, decides whether the projected symbols are falling in region A or region B (see Figure 4). If the symbols fall in region A, then it will decide that 1 was transmitted. It they fall in region B, the decision will be in favor of ‘0’.

For deriving the performance of the receiver, the decision process made by the comparator is applied to the underlying distribution model (Figure 5). The symbols projected on the axis will follow a Gaussian distribution. The threshold for decision is set to T=0. A received bit is in error, if the transmitted bit is ‘0’ & the decision output is ‘1’ and if the transmitted bit is ‘1’ & the decision output is ‘0’.

This is expressed in terms of probability of error as,


Or equivalently,

By applying Bayes Theorem↗, the above equation is expressed in terms of conditional probabilities as given below,


Since a-prior probabilities are equal P(0T)= P(1T) =0.5, the equation can be re-written as

Intuitively, the integrals represent the area of shaded curves as shown in Figure 6. From the previous article, we know that the area of the shaded region is given by Q function.

Figure 6a, 6b: Calculating Error Probability

Similarly,

From (4), (6), (7) and (8),


For BPSK, since Es=Eb, the probability of symbol error (Ps) and the probability of bit error (Pb) are same. Therefore, expressing the Ps and Pb in terms of Q function and also in terms of complementary error function :


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Reference

[1] Nguyen & Shwedyk, “A First course in Digital Communications”, Cambridge University Press, 1st edition.↗

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