ELE520: Machine Learning – Multivariate Probability Density Function – Engineering Assignment Help

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Internal Code: 3IIE
Engineering Assignment Help
Task:
Problem 1
Assume that the underlying a priori probabilities and class conditional probability density functions from problem 2, exercise 2 is unknown. However, we have access to measurements so that we have the following samples from the two categories (also illustrated in figure 1).

Engineering

 Assuming a gaussian distribution, you are supposed to use a parametric ap- proach for formulating the Bayes classifier from problem 2 in exercise 2, based on the two data sets. Apply the maximum-likelihood (ML) method to esti- mate the required functions. (The expression will look ugly, so do not exhaust yoursel trying to simplify it.)

 Compare the decision border with the one computed in problem 2 in exercise 2. How are the two estimated density functions oriented in relation to each other and in relation to the true density functions? (Do not perform eigenanalysis, base your answer on observing the nature of the expression for the decision boundary.)

How can you make the decision border correspond better to the one found in 
problem 2, exercise 2.

Engineering
Problem 2
Using the data set from the previous problem, you are supposed to classify the feature vector x = (2.5 2.0)
T
. Use the following classifiers:

The Bayes classifier from the previous problem.

A Parzen-window classifier. Use a gaussian window function so that

Engineering

A Parzen-window classifier as in the previous subtask, but this time let h
1
= 5.
Compare with the results from the previous subtask and explain what has happened.

A k
N
-nearest neighbourhood classifier where k
N
= 1.

A k
N
-nearest neighbourhood classifier where k
N
= 3.

Problem 3
Derive the maximum-likelihood-estimate for

Engineering

for the case where both ? og ? in the multivariate probability density function

Engineering

Laboratory exercise

In this problem the purpose is to visualise the use of parametric and non parametric estimation techniques for the minimum error rate classifiers from theoretical exercise 3. Training data are stored in the pickle file lab3.p and can be downloaded from CANVAS.
For both the estimation approaches it might be useful to apply norm2D to compute the estimated denisty function values. (This demands some considerations for the Parzen-technique). For the nearest neighbourhood method you have to make a new function.
Problem 1

Estimate
?
i
from each data set
/
i
for i = 1,2. (Hint: Use the numpy command 
mean
.)

Estimate ?
i
from each data set
/
i
for i = 1,2.(Hint: Use the numpy command 
cov
.)

Estimate the discriminant function (scaled probability density function) for class ?
1
and plot this with red surface color (
facecolor=’r’;
). Define 25
points of computation along each axis so that you will be using 25
2
points of computation.

Estimate the discriminant function for class ?
2
and plot it with blue surface
(
facecolor=’b’;
) so that the two functions are shown in the same figure.

Identify the decision border and decision regions. Compare with the discrimi
nant functions that were plotted in laboratory exercise 2.

Repeat subtaske c-e for the Parzen classifier from problem 2b in theoretical 
exercise 3.

Repeat subtaske c-e for the Parzen classifier from problem 2c in theoretical 
exercise 3..

Repeat subtaske c-e for the k
N
-nearest neighbour classifier from subtask 2d.

Repeat subtaske c-e for the k
N
-nearest neighbour classifier from subtask 2e.

Add functionality so that the figure display the a posteriori probability for the 
two classes

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