This document summarizes topics that will be covered by a group of EXTC students in 2022-2023 for their Random Signals Analysis course. The topics include: ergodicity, Markov chains, and statistical learning and its applications. Ergodicity concepts such as mean ergodic processes and correlation ergodic processes will be discussed. Markov chains will be defined and different classifications will be examined. Statistical learning techniques like simple and multiple regression analysis will also be explored, along with comparing correlation and regression.

1. RANDOM SIGNALS
ANALYSIS
THIRD YEAR EXTC 2022-2023
GROUP MEMBERS:-
DARSHAN AHER
ADNAN SHAIKH
RUCHI SHENDAGE

2. TOPICS :-
•ERGODICITY
•MARKOV CHAINS
•STATISTICAL LEARNING AND ITS APPLICATIONS

3. ERGODICITY :

4. MEAN ERGODIC PROCESS :

5. CORRELATION IN ERGODIC PROCESS :
The stationary process {X(t)} is said to be correlation ergodic (or ergodic in the correlation), if the
process {Y(t)} is mean-ergodic, where
Y(t) = X(t+ τ ) . X(t)
That is the Stationary Process {X(t)} is correlation ergodic, if
Thus to prove that the given Random Process {X(t)} is correlation- ergodic, we have to prove that
1. {X(t)} is stationary and
2.

6. Ergodic Power Spectral Density
Function :

7. Properties of Power Spectral Density :

8. 3. The Spectral Density Function of a real Random Process is an even Function .
4. The Spectral Density of a Process {X(t)}, real or complex, is a real function of w and non-negative.
5. The Spectral Density and autocorrelation function of a real WSS process form a Fourier cosine
transform pair .

9. CROSS SPECTRAL DENSITY FUNCTION :

10. PROPERTIES OF CROSS SPECTRAL DENSITY :

11. MARKOV CHAIN

12. Definition :- If, for all n,
P[ Xn = an/Xn-1 = an-1,Xn-2=an-2…..,X0=0 ]
= P [ Xn = an/Xn = an-1, then process [Xn], n = 0,1,2…. Is called a Markov Chain.
States of Markov Chain :-
(a1,a2,an,….) are called the States of The Markov Chain.
Transition Probability Matrix :-
When the system changes from ith stage to jth stage with probability Pij , the
probability is called transition probability .

13. CLASSIFICATION ON MARKOV CHAIN :

17. Statistical Learning and Its
Applications :

18. UTILITY OF REGRESSION
 Degree & Nature of relationship
 Estimation of relationship
 Prediction
 Useful in Economic & Business
Research

19. DIFFERENCE BETWEEN CORRELATION & REGRESSION
 Degree & Nature of Relationship
⚫ Correlation is a measure of degree of relationship between X & Y
⚫ Regression studies the nature of relationship between the
variables so that one may be able to predict the value of one
variable on the basis of another.
 Cause & Effect Relationship
⚫ Correlation does not always assume cause and effect
relationship between two variables.
⚫ Regression clearly expresses the cause and effect relationship between
two variables. The independent variable is the cause and dependent
variable is effect.

20.  Prediction
⚫ Correlation doesn’t help in making predictions
⚫ Regression enable us to make predictions using regression line
 Symmetric
⚫ Correlation coefficients are symmetrical i.e. rxy = ryx.
⚫ Regression coefficients are not symmetrical i.e. bxy ≠ byx.
 Origin & Scale
⚫ Correlation is independent of the change of origin and scale
⚫ Regression coefficient is independent of change of origin
but not of scale

21. TYPES OF REGRESSION ANALYSIS
 Simple & Multiple Regression
 Linear & Non Linear Regression
 Partial & Total Regression

22. SIMPLE LINEAR REGRESSION

23. REGRESSION LINES
 The regression line shows the average relationship between two variables.
It is also called Line of Best Fit.
 If two variables X & Y are given, then there are two regression lines:
⚫ Regression Line of X on Y
⚫ Regression Line of Y on X
 Nature of Regression Lines
⚫ If r = ±1, then the two regression lines are coincident.
⚫ If r = 0, then the two regression lines intersect each other at
90°
⚫ The nearer the regression lines are to each other, the greater will be the degree of
correlation.
⚫ If regression lines rise from left to right upward, then correlation is positive.

24. REGRESSION EQUATIONS

25. REGRESSION COEFFICIENTS

26. PROPERTIES OF REGRESSION COEFFICIENTS

27. OBTAINING REGRESSION EQUATIONS

28. REGRESSION EQUATIONS IN INDIVIDUAL SERIES
USING NORMAL EQUATIONS

29. REGRESSION EQUATIONS USING REGRESSION
COEFFICIENTS

30. REGRESSION EQUATIONS USING REGRESSION COEFFICIENTS (USING ACTUAL
VALUES)
REGRESSION EQUATIONS USING REGRESSION COEFFICIENTS (USING DEVIATIONS FROM
ACTUAL VALUES)

31. REGRESSION EQUATIONS USING REGRESSION COEFFICIENTS (USING DEVIATIONS
FROM ASSUMED MEAN)
REGRESSION EQUATIONS USING REGRESSION COEFFICIENTS (USING STANDARD DEVIATIONS)

32. SHORTCUT METHOD OF CHECKING REGRESSION EQUATIONS

33. STANDARD ERROR OF ESTIMATE

34. THANKYOU