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Mathematics & Discrete
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The following shows 5 transactions for these two customers. Each transaction contains three kinds of information: customer ID (e.g., X and Y),

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ANSWER ALL QUESTIONS

Q1 [20 Marks]

  • In general, we have a number of For illustration, we are given two customers, namely X and

  1. The following shows 5 transactions for these two customers. Each transaction contains three kinds of information: (1) customer ID (e.g., X and Y), (2) the time that this transaction occurred, and (3) all the items involved in this transaction.

 

Customer X, time 1, items A, B, C Customer Y, time 2, items A, F Customer X, time 3, items D, E Customer X, time 4, item G Customer Y, time 5, items D, E, G

For example, the first transaction corresponds to that customer X bought item A, item B and item C at time 1, while the last transaction corresponds to that customer Y bought item D, item E and item G at time 5.

 

A sequence is defined to be a series of itemsets in form of <S1, S2, S3, …, Sm> where Si is an itemset for i = 1, 2, …, m. The above transactions can be transformed into two sequences as follows.

 

X: <{A, B, C}, {D, E}, {G}>

Y: <{A, F}, {D, E, G}>

 

After this transformation, each customer is associated with a sequence.

 

Given a sequence S in form of <S1, S2, S3, …, Sm> and another sequence S’ in form of <S1’, S2’, S3’, …, Sn’> , S is said to be a subsequence of S’ if m £ n and there exist m integers, namely i1, i2, …, im, such that (i) 1£i1<i2< …< im£n, and (2) Sj Í Sij’ for j = 1, 2, …, m. If S is a subsequence of S’, then S’ is defined to be a super-sequence of S.

The support of a sequence S is defined to be the total number of customers which sequences are super- sequences of S.

Given a positive integer k, a sequence in form of <S1, S2, S3, …, Sm> is said to be a k-sequence if

å |Si|=k.

i=1

Can the Apriori algorithm be adapted to mining all k-sequences with support at least 2 where k = 2, 3,  4, …. ? If yes, please write down the proposed method using the concept of the Apriori algorithm and illustrate your algorithm with the above example. If no, please explain the reason.

 

  • We want to study the same problem setting described in (a). However, each customer is associated to one binary attribute called “Rich” to indicate whether this customer is rich or not. There are only 2 possible values in this attribute, namely “Yes” and “No”. In our example, customer X could have “Yes” in attribute “Rich” and customer Y could have “No” in attribute “Rich”.

 

Given a k-sequence s and a value v in attribute “Rich”, the support of a sequence S with respect to value v is defined to be the total number of customers which sequences are super-sequences of S and are associated with value v in attribute “Rich”. The important ratio of s is defined to be the support of s with respect to value “Yes” divided by the support of s with respect to value “No”.

 

Can the Apriori algorithm be adapted to mining all k-sequences with important ratio at least 2 and the support at least 1 where k = 2, 3, 4, ….? If yes, please write down the proposed method using the concept of the Apriori algorithm and illustrate your algorithm with the above example. If no, please explain the reason.

 

Note that when we compute the important ratio, if we encounter a division of a non-zero number by zero, we could regard it as a positive infinity value.

 

Q2 [20 Marks]

 

Given a positive integer K, we denote SK to be a set of K-itemsets with support at least 1.

Given a positive integer K and a positive integer l, we define a set SK, l which is a subset of SK such that  each K-itemset in SK, l has its support at least sl where sl is the l-th greatest value in the multi-set of the supports of all K-itemsets in SK. For example, the second greatest value in a multi-set of {4, 4, 3, 2} is 4 while the second greatest value of another multi-set of {4, 3, 3, 2} is 3.

We are given six items, namely A, B, C, D, E and F. Suppose l is fixed and is set to 2.

We want to find SK, l for K = 1, 2 and 3.

 

The following shows four transactions with six items. Each row corresponds to a transaction where 1 corresponds to a presence of an item and 0 corresponds to an absence.

A

B

C

D

E

F

0

0

1

1

0

0

0

1

0

0

1

1

1

0

1

1

0

0

1

0

1

1

0

0

 

  • (i) What is S1, 2?

    • What is S2, 2?

    • What is S3, 2?

  • Can algorithm FP-growth be adapted to finding S1, 2, S2, 2 and S3, 2. If yes, please write down how toadapt algorithm FP-growth and illustrate the adapted algorithm with the above example. If no,  please explain the reason.

  • There are two parameters of finding SK, l. They are K and l. In the traditional problem of findingfrequent itemsets, we need to provide only one parameter, a support

 

It seems that it is troublesome to set one more parameter in the problem of finding SK, l (compared with the traditional frequent itemset mining you learnt). What are the advantages of the problem of finding SK, l compared with the traditional problem?

 

Q3 [20 Marks]

 

  • One disadvantage of method k-means is that k (i.e., the number of clusters) should be pre-determined. One may suggest the following method to determine parameter

    • Step 1: Set variable e0 to ¥

    • Step 2: Set k to 1 initially

    • Step 3: Run the original k-means method and obtain k cluster centers (or means)

    • Step 4: Set variable ek to the sum of the distances between points and their closest cluster centers(according to the k cluster centers found).

    • Step 5: If ek converges (i.e., (ek-1 – ek) is equal to 0 or an extremely small number), then return k.Otherwise, increment k by 1 and repeat Step 3 to Step

Can the above method determine a good value for k (i.e., the number of clusters)? Please explain.

If your answer is no, please also give an algorithm to determine a good value for k and explain why it is better than the above method.

 

  • In a typical setting for k-means, we calculate the distance between a point and a cluster by computing the Euclidean distance between a point and the center (or mean) of the

 

 

Q4 [20 Marks]

 

  • Consider the following eight two-dimensional data points:

 

x1:(20, 15), x2: (8, 15), x3: (20, 17), x4: (8, 19), x5: (23, 18), x6: (6, 12), x7: (15, 6), x8: (15, 35)

 

  • Assume that we adopt the Euclidean distance metric as the distance between any two points. Please write a matrix where its entries correspond to the pairwise distances between any two points.

  • Please use  the  agglomerative  approach  to  group  these  points  with   centroid     Draw the corresponding dendrogram for the clustering. You are required to specify the distance metric in the dendrogram.

 

  • According to (a), we have eight data points. Suppose that we have one more data point. Totally, we have nine data points. Is it possible for you to design an algorithm so that the dendrogram obtained in (a) could be updated efficiently? If yes, please give a step-by-step algorithm. Otherwise, please elaborate in

 

Q5 [20 Marks]

 

In the class, we learnt algorithm DBSCAN with 4 principles on a given dataset.

 

  • Please write down a pseudo-code in order to perform clustering according to these 4

  • Suppose that the dataset is updated from time to

    • Please give a pseudo-code in order to perform clustering according to these 4 principles on this updating dataset. It is expected that the algorithm should return clustering results based on both the information stored based on the past data together with the new

    • Please analyze the time complexity of the

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