Lecture 10: Inference in Gaussian Linear State Space Models:
Filtering & Smoothing
John MacLaren Walsh, Ph.D.
March 11, 2015
References
Inference in Hidden Markov Models, Olivier Cappe, Eric Moulines, Tobias Ryden. Springer, 2005. See
Chapter 5.2.
Patte
Homework #7 Solution
ECE S523
1. Let 1 be the decision based on feature 1 and let 2 be the decision based on feature 2. k, k = 1,
2, takes the value 1 or 0 depending on whether H1 or Ho is picked, and let p(Ho) = p(H1). Let
1
= ( ) be the vector whose el
ECE S523
Homework # 7
1. Let 1 be the decision based on feature 1 and let 2 be the decision based on feature 2. k,
k = 1, 2, takes the value 1 or 0 depending on whether H1 or Ho is picked, and let p(Ho) =
1
p(H1). Let = ( ) be the vector whose elements ar
ECE 523
Homework 5
Spring 2015
2
1. Let =
=0 + , where is (0, ), = 1,2, , , is white noise.
a. Assume that 0 , 1 , . . , and 2 are known, how would you determine the best
, 0 ?
b. Find the MLE for 0 , 1 , . . , and 2.
c. Find the MMSE estimates for 0 , 1
Homework # 4
1. You are given a set of n independent identically distributed random vectors ri = (xi, yi), i = 1,
2, ., n from a normal bivariate distribution (0, 0, 1, 1, ).
(a) Find the minimal sufficient statistics T(r1, r2, , rn) associated with .
(b)
Homework # 3
1. If = (1 , 2 , . , ) are i.i.d. distributed with (| ) = 2 x , 0, > 0.
a) Find the minimum sufficient statistics associated with .
b) Is the minimum sufficient statistics complete? Why?
c) Find the MVUE of . Is it unique?
2. Show that if = (
Homework # 2
1. If x1, x2, ., xN are i.i.d. Poisson distributed with (| ) = , 0, > 0.
!
a) Show that (| x1 ) = (x1 = ), = 0, 1, is unbiased estimator for (| ) based on
one observation x1, where (x1 = ) is an indicator function taking the value 1 if x1 =
ECE S-523
Detection and Estimation
Spring 2014
Homework # 1
1. For a two class problem two-dimensional problem let the class conditional
0
1
distribution p(x|i) ~ (i,), i = 1,2, where
( ), 1 , 2 ,
1
2
1 0
. Assume that p(1)= p(2)=0.5.
0 4
a) Find th
ECES 690/435 - Winter 2015
Instructor: Matthew Stamm
Drexel University
Assignment 4 - Part 1
This assignment is due by 11:59 p.m. on Thursday, March 5. All reports should be submitted as PDFs. Only one
report must be submitted per team, but all team membe
ECES 690/435 - Winter 2015
Instructor: Matthew Stamm
Drexel University
Assignment 4 - Part 2
This assignment is due by 11:59 p.m. on Thursday, March 5. All reports should be submitted as PDFs. Only one
report must be submitted per team, but all team membe
ECES 690 Homework 3
Sagar Patel , Suman Satish
February 20, 2015
1
1
Part 1
The code for watermarking the image using Spread Spectrum Watermarking
has been appended in the end.
2
Part 2
The code for extracting the watermark from an already watermarked ima
ECES 690/435 - Winter 2015
Instructor: Matthew Stamm
Drexel University
Assignment # 3, Part 1
This assignment is due by 11:59 p.m. on Thursday, Feb 19. All reports should be submitted as PDFs. Only one
report must be submitted per team, but all team membe
ECES 690/435 - Winter 2015
Instructor: Matthew Stamm
Drexel University
Assignment # 2, Part 1
This assignment is due by 11:59 p.m. on Thursday, Feb 12. All reports should be submitted as PDFs. Only one
report must be submitted per team, but all team membe
ECES 690 Homework 1
Sagar Patel , Suman Satish
January 30, 2015
1
1
1.1
Part 1
Subsection 1-A
A MATLAB function that performs gamma correction on a greyscale digital
image is attached in the appendix
Figure 1: Gamma correction from top left: original imag
ECES 690/435 - Winter 2015
Instructor: Matthew Stamm
Drexel University
Assignment # 1
This assignment is due by 11:59 p.m. on Wednesday, January 29. All reports should be submitted as PDFs. Only
one report must be submitted per team, but all team members
Week 1: Dening Stochastic Processes, Stationarity, & Wide Sense
Stationarity
John MacLaren Walsh, Ph.D.
1
What is a Stochastic Process? How do I Dene it?
A discrete time stochastic process can be thought of as an innite sequence of random variables X : Z
Week 2: MMSE Linear Prediction & Filtering for Wide Sense
Stationary Random Processes
John MacLaren Walsh, Ph.D.
1
Causal Linear MMSE Prediction
Suppose we wish to predict the value of a zero mean DT-WSSRP X[n] based on the the values from at least
d samp
Week 4: Discrete Time Markov Chains, II:
State Classication, Stationary Distributions, & Limiting Behavior
John MacLaren Walsh, Ph.D.
1
References
Markov chains are a widely taught subject, and hence there are a wide variety of texts. A few that cover
thi
Week 8: Renewal Processes, Stopping Times, Walds Equation
John MacLaren Walsh, Ph.D.
1
References
R. Gallager, Stochastic Processes: Theory for Applications, Cambridge University Press, 2014.
A. Leon-Garcia, Probability and Random Processes for Electric
Week 6: Continuous Time Homogenous Markov Chains & Basic
Queuing
John MacLaren Walsh, Ph.D.
1
References
A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd ed., Addison Wesley
Longman, 1994.
A. Papoulis, Probability, Random
ECE S681 Homework # 1
A weaving machine capable of generating horizontal streak defects (of one pixel width) at random
locations needs to be tested using an FIR sensor that ideally sees the defects as bright spots (value of 1)
and the fabric as black spot
ECE-S 681 Homework #2 Solution
Part 1: Theoretical Questions
1. If y = f(x) = m x + d is the nonparametric explicit linear polynomial representing a curve, and
(x(t), y(t), with x(t) = a1 t + b1 and y(t) = a2 t + b2, is the parametric equivalent linear
po