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Mathematics & Discrete
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Consider the density-based subspace clustering. The size of a subspace is defined to be the total number of dimensions for this subspace

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Q1 [20 Marks]

 

Consider the density-based subspace clustering. The size of a subspace is defined to be the total number of dimensions for this subspace. For example, subspace {A, B} is of size 2. For each single dimension, the number of grid units is fixed to a constant c where c is a positive integer greater than 1.

  • In class, we learnt that the major idea in the KL-transform is to transform the original coordinate system to a new coordinate system such that we could find clusters in subspaces from the new coordinate Suppose that we use the KL-transform to transform all data points from the original coordinate system to a new coordinate system (without using Step 6 of the KL-transform (i.e., choosing a subset of attribute values)). Then, all points are now represented in the new coordinate system. Based on the new coordinate system, we adopt the density-based subspace clustering to find clusters in some subspaces. Is it always true that the total number of grid units involved in all clusters based on the new coordinate system is smaller than that based on the original coordinate system? If yes, please give some justifications without any formal proof. If no, similarly, please give some counter examples for illustration.

  • When the size of the subspace is larger, it is less likely that a grid unit with respect to the subspace is dense. Please explain

  • In order to overcome the weakness described in (b), instead of setting a fixed density threshold for the subspace of any size, we use a smaller density threshold for the subspace of larger size. Specifically, let Ti be the density threshold for the subspace of size i. If i < j, then Ti > Tj. Let Condition 1 be “Ti > Tj forany i < j”.

Let Condition 2 be “for any i and j, Ti = Tj”. We know that if Condition 2 is satisfied, then the original Apriori-like algorithm studied in class can find all subspaces containing dense units.

  • Under Condition 1, is it always true that we can still adopt the Apriori-like algorithm? If yes, please describe how to adopt the algorithm. Otherwise, please give reasons why it cannot be

  • Suppose that we modify Condition 1 to the following form. Let Condition 1 be “Ti = cTi+1 for eachpositive integer i”. Assume that we adopt this new form of Condition 1. Under this new form of Condition 1, is it always true that we can still adopt the Apriori-like algorithm? If yes, please describe how to adopt the algorithm. Otherwise, please give reasons why it cannot be

 

Q2 [20 Marks]

  • Consider a set P containing the following four 2-dimensional data points. a:(6, 6), b:(8, 8), c:(5, 9), d:(9, 5)

We can make use of the KL-Transform to find a transformed subspace containing a cluster. Let L be the total number of dimensions in the original space and K be the total number of dimensions in the projected subspace. Please illustrate the KL-transform technique with the above example when L=2 and K=1.

 

  • Consider a set Q containing the following four 2-dimensional data points. e:(5, 5), f:(9, 9), g:(3, 11), h:(11, 3)

    • Let p = (xp, yp) be a point in P and q = (xq, yq) be a point in In fact, we could express xq in a linear form involving xp such that xq = α . xp + β where α and β are 2 real numbers. Similarly, we could express yq in the same linear form involving yp. Please write down the values of α and β.

    • Similar to Part (a), we want to make use of the KL-Transform to find a transformed subspace containing a cluster for the set Q where L = 2 and K = 1. One “straightforward” or “naïve” method is to use the same method in Part (a) to obtain the answer. Is it possible to make use of the result in Part (a) and the result in Part (b)(i) to obtain the answer very quickly? If yes, please explain briefly and give the answer. There is no need to give a formal proof. A brief description it accepted. If no, please give an explanation briefly. In this case, derive the answer by using the method in Part (a).

 

  • Consider Part (a). It is independent of Part (b). In Part (a), we know that there are 4

Suppose that we have 4 additional points which are identical to the original 4 points. That is, we have the following 4 additional points. Totally, we have 8 data points.

 

(6, 6), (8, 8), (5, 9), (9, 5)

 

One “straightforward” or “naïve” method is to use the same method in Part (a) to obtain the answer. Is it possible to make use of the result in Part (a) to obtain the answer very quickly? If yes, please explain briefly and give the answer. There is no need to give a formal proof. A brief description it accepted. If no, please give an explanation briefly. In this case, derive the answer by using the method in Part (a).

 

  • Consider two random variables X and Y with the following probabilistic

 

X Y

1

2

3

1

0

1/8

1/8

2

1/2

0

1/8

3

1/8

0

0

 

  • Calculate the conditional entropy of H(X|Y) by using the original definition of the conditional entropy.

  • Calculate H(X|Y) as

- åxÎA åyÎB p(x, y) log p(x|y) where A = {1, 2, 3} and B = {1, 2, 3}.

 

Q3 [20 Marks]

The following shows a history of PhD students with their numbers of published papers, their ages and their majors. We also indicate whether they become professors or not after their PhD graduation in the last column. Note that the first column “No.” is for us to refer the record number only.

No.

NoOfPapers

Age

Major

Become_Professor

1

enough

young

ComputerScience

yes

2

many

young

ComputerScience

yes

3

many

old

CivilEngineering

yes

4

many

old

DataScience

yes

5

few

young

CivilEngineering

no

6

many

young

ComputerScience

no

7

few

old

DataScience

no

8

few

old

ComputerScience

no

 

  • We want to train a C4.5 decision tree classifier to predict whether a PhD student will become a professor or not. We define the value of attribute Become_Professor to be the label of a

    • Please find a C4.5 decision tree according to the above example. In the decision tree, whenever we process (1) a node containing at least 80% records with the same label or (2) a node containing at most 2 records, we stop to process this node for

    • Consider a young PhD student majoring in computer science who published many papers. Please estimate the probability that this PhD student will become a

  • Let X be the set of attributes involved in the decision tree found in Part (a). Person A said that we just need to consider all attributes in X only to determine whether a student will become a professor. Person B said that we should also consider attributes outside X (in addition to attributes in X) to determine whether a student will become a

    • Please give a possible reason why Person A said in this

    • Please give a possible reason why Person B said in this

    • Which Person (Person A or Person B) is more unreasonable in general?

  • What is the difference between the 5 decision tree and the ID3 decision tree? Why is there a difference?

 

Q4 [20 Marks]

Suppose that there is a new patient. We know that

  • he has acute pancreatitis

  • he has pneumonia

  • his result of white blood cell is low

We would like to know whether he is likely to have systemic inflammation reaction.

Acute Pancreatitis

Pneumonia

White Blood Cell

Systemic Inflammation Reaction

Yes

Yes

Low

?

  • Please use Bayesian Belief Network classifier with the use of Bayesian Belief Network to predict whether he is likely to have systemic inflammation

  • Although Bayesian Belief Network classifier does not have an independent assumption among all attributes (compared with the naïve Bayesian classifier), what are the disadvantages of this classifier?

 

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