ELE520: Machine Learning – Electrical Engineering and Computer Science Assignment Help

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Internal Code: 3AJDG
Machine Learning – Electrical Engineering and Computer Science Assignment Help
Task:
Problem 1
We shall design a pattern recognition system to classify 2-dimensional feature vectors {x} to class ?
1
(’×’), or class ?
2
(’?’). As a basis for this work, we have performed
measurements so that 4 feature vectors are available as training vectors: 

a) Plot the training vectors in the x
1
x
2
–plane translated to the augmented feature

space which will be used in the following. estimate the a priori probabilities.
b) We want to design a discriminant function on the form:

where    
This is the same data set as in the example collec-
tion where straightforward linear regression was attempted and shown not to produce any result meaningful for discrimination.
Use the LS-method to determine ?. Determine and draw the decision boundary defined by ?.
c) What is the effect on the decision boundary of changing y
4
to ?0.5.
d) Use the LMS-method to determine ?. Choose the starting point ?
(1)
= (1 1 1)
t
. Set the threshold value to ? = 1, and let ? = 0.5. Remember that the learning rate is updated according to ?(i) = ?/i. Compute ?
(i)
for i = 2,3,4, … until convergence and sketch the decision boundary defined by ?. Compare the resulting decision boundary with the one you found in subtask 
Trygve Eftestøl, Professor
Faculty of Science and Technology University of Stavanger Telephone: +47 51 83 10 00. Department of Electrical and Computer Engineering N-4036 Stavanger. Telefax: +47 51 83 17 50. E-mail: [email protected] Kjølv Egelands hus www.ux.his.no/ ?trygve-e
Problem 2
The problem of defining a generalised linear discriminant function to discriminante the vectors in a labelled two-class problem can be expressed
X? = y (1)
where X is the N ×
ˆ
l matrix
1
where the nth row is the vector x
n
T
and y be the
vector y = (y
1
,…,y
N
)
T
.
a) Explain why this equation is not generally solvable.
b) Show that the vector ? giving the best approximation in the sense of the minimum squared error
 between the right and left side in the equation is given by

c) Find an expression for the quadratic distance using the solution from the
previous task.
d) Show that the problem is linearly separable ifwhere I is the identity matrix.
Problem 3
Consider the data sets with samples x
i
 

for class ?
2
. We want to find the discriminant function g(x) = ?
T
x that solves the
inequalities ?
T
x
i
> 0.
The batch perceptron algorithm is given:
Algorithm 3. (Batch Perceptron)

a) Augment the data and determine the labels for each sample.
b) Plot the lines defined by ?
T
x
i
= 0 for all the samples.
c) Indicate the positive and negative sides of the lines and identify the solution
region.
d) Apply the algorithm letting the initial values be ?
(1)
=0,?(1) = 1, criterion  
? = 0. Let ?(i) = 1.
e) As above, but let let ?(i)=1/i.

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