AEM 5451/EE 5251 PROBLEM SET #2 solution

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Problem 1
Do problem 2.5 in your text.
Problem 2
h) Are X and Y independent? Show this by determining the marginal distributions
fX(x) and fy(y).
Problem 3
Number of Residents Number of Residents with
No Felony Conviction One or More Felony Convictions
Female Residents 359, 881 172, 742
Male Residents 340, 454 40, 854
You are visiting an island that is a prison colony for political prisoners from the dystopian
planet Tralfamador. When a political prisoner receives a guilty verdict in court, (s)he is
declared a felon and sent to the island along with their immediate family members. The
felony conviction statistics for the current residents of the island are given the table above.
Consider the following outcomes for a resident of the island picked at random: (1) M = male
resident;(2) F = female resident; (3) C = convicted felon; (4) I = no felony record. Using
the data given and assuming that relative frequencies are equal to probabilities, calculate the
following: P(M), P(F), P(C), P(I), P(M ∩I), P(F ∩I), P(M ∩ C), P(F ∩ C). Now using
frequencies computed from the given data.
a) What is the probability that a male resident of the island picked at random will be a
convicted felon?
b) What is the probability that a female resident of the island picked at random will be
a convicted felon?
c) What is the probability that an island resident picked at random is female, given that
that they are a convicted felon?
d) What is the probability that an island resident picked at random is male, given that
they are a convicted felon?
Problem 4
Do problem 3.9 in your text. This is the problem we kicked-off the course with and now you
have the tools to find its solution. HINT: Treat it as a recursive least squares problem.
Problem 5
Figure 1: A system with an unknown mathematical model.
This problem demonstrates the utility and power of a simple, least squares estimator.
You will use least squares to determine the mathematical model of an unknown dynamical
system. The system can be an unknown electrical circuit, for example, which you can
stimulate and then record its output (e.g., voltage-in-voltage out). Or alternately, it can
be a vehicle whose equations of motion your do not know. So your apply control inputs
(e.g., turn the vehicle right and left rapidly; accelerate and decelerate it, etc) and observe its
kinematic state (e.g., position, velocity, acceleration, etc). Such problems are called system
identification (or simply sysid) and they are a small part of what we know as machine
learning today.
Download the data file inoutdata.txt found on the assignment page. This data contains
the output from the unknown (or “blackbox”) system shown in Figure 1. The first column
of the data is the time tk, the second column is the input uk and the third column is the
output yek (that is, yk when vk = 0). The system dynamics and observation models are:
αk = F αk−1 + Guk−1
yk = ye+ vk = Cαk + vk
2
Suppose you know that the matrices F, G and H have the following form:
F =

−a1 1
−a2 0

, G =

b1
b2

, C =

1 0
Your job is to use least squares to estimate (or identify) the parameters or entries of F and
G. That is, generate an optimal estimate of:
x =

a1 a2 b1 b2
T
You will proceed to do this as follows:
a) Write out the measurement equation for your estimator in the form
ye = Hx
where ye is the error-free sensor measurement (i.e., case when v = 0). Clearly identifying
the entries in ye and H. This is by far the most challenging part of the problem. HINT:
Try to relate yk to yk−1, yk−2, . . . , uk−1, uk−2, . . . and so using an autoregrsive moving
average (ARMA) model. If you do not know what an ARMA model is, look it up.1
b) Estimate the state vector using normal least squares.
c) Note that error-free yek is not available to you in practice. Instead you will use some
sensor to measure an error corrupted version denoted yk. Use the MATLAB command
randn to simulate the sensing process as follows:
yk = yek + 0.1 ∗ randn(N, 1)
where N is the number of data points in yek you are using for the estimation process.
d) Use the history of yek you generated and uk to generate an estimate of x.
e) Repeat the estimation process but now use a less noisy sensor whose output yk is given
by:
yk = yek + +0.001 ∗ randn(N, 1)
Does the value of x change appreciably?
f) Comment on how you would validate the accuracy of the estimate you generated.
1As of the moment I was writing this homework set (September 5, 2020), the Wikipedia Article on ARMA
is rather verbose and hard to follow. You can read it to learn what an ARMA model is but I would suggest
looking elsewhere to learn about ARMA.
3